1 Introduction

The theory of harmonic morphisms was initiated by Jacobi, in the 3-dimensional Euclidean geometry, with his famous paper [8] from 1848. In the late 1970s this was then generalised to Riemannian geometry in [3] and [7] by Fuglede and Ishihara, independently. This has led to a vibrant development which can be traced back in the standard reference [2], by Baird and Wood, and the regularly updated online bibliography [6], maintained by the second author. The following result of Baird and Eells gives the theory of harmonic morphisms, with values in a surface, a strong geometric flavour.

Theorem 1.1

[1] Let \(\phi :(M^m,g)\rightarrow (N^2,h)\) be a horizontally conformal submersion from a Riemannian manifold to a surface. Then \(\phi\) is harmonic if and only if \(\phi\) has minimal fibres.

In their work [5], the authors classify the 4-dimensional Riemannian Lie groups (G, g) equipped with a 2-dimensional, conformal and left-invariant foliation \(\mathcal {F}\) with minimal leaves. Such a foliation is locally given by a submersive harmonic morphism into a surface. For such a Riemannian Lie group there exist natural almost Hermitian structures J adapted to the foliation structure.

The purpose of this work is to classify these structures which are almost Kähler \((\mathcal {A}\mathcal {K})\), integrable \((\mathcal {I})\) or Kähler \((\mathcal {K})\), see [4]. Our classification result is the following.

Theorem 1.2

Let (G, g, J) be a 4-dimensional almost Hermitian Lie group equipped with a 2-dimensional, conformal and left-invariant foliation \(\mathcal {F}\) with minimal leaves. If a natural almost Hermitian structure J, adapted to the foliation \(\mathcal {F}\), is almost Kähler \((\mathcal {A}\mathcal {K})\), integrable \((\mathcal {I})\) or Kähler \((\mathcal {K})\), then the corresponding Lie algebra \(\mathfrak {g}_k^{\mathcal {A}\mathcal {K}}\), \(\mathfrak {g}_k^{\mathcal {I}}\) or \(\mathfrak {g}_k^{\mathcal {K}}\) of G is one of those given below \((k=1,2\dots ,20)\).

2 Two dimensional conformal foliations \(\mathcal {F}\) on \((G^4,g)\)

Let (M, g) be a Riemannian manifold, \(\mathcal {V}\) be an integrable distribution on M and denote by \(\mathcal {H}\) its orthogonal complementary distribution. As customary, we shall also denote by \(\mathcal {V}\) and \(\mathcal {H}\) the orthogonal projections onto the corresponding subbundles of the tangent bundle TM of M and denote by \(\mathcal {F}\) the foliation tangent to \(\mathcal {V}\). Then the second fundamental form for \(\mathcal {V}\) is given by

$$\begin{aligned} B^\mathcal {V}(Z,W)=\frac{1}{2}\,\mathcal {H}({\nabla _{Z}}W+{\nabla _{W}}Z)\qquad (Z,W\in \mathcal {V}), \end{aligned}$$

while the second fundamental form for \(\mathcal {H}\) satisfies

$$\begin{aligned} B^\mathcal {H}(X,Y)=\frac{1}{2}\,\mathcal {V}({\nabla _{X}}Y+{\nabla _{Y}}X)\qquad (X,Y\in \mathcal {H}). \end{aligned}$$

The foliation \(\mathcal {F}\) tangent to \(\mathcal {V}\) is said to be conformal if there exists a vector field \(V\in \mathcal {V}\) such that the second fundamental form \(B^\mathcal {H}\) satisfies

$$B^\mathcal {H}=g\otimes V$$

and \(\mathcal {F}\) is said to be Riemannian if \(V=0\). Furthermore, \(\mathcal {F}\) is said to be minimal if \(\text {trace}\ B^\mathcal {V}=0\) and totally geodesic if \(B^\mathcal {V}=0\). This is equivalent to the leaves of \(\mathcal {F}\) being minimal or totally geodesic submanifolds of M, respectively.

It is well-known that the fibres of a horizontally conformal map (resp. Riemannian submersion) give rise to a conformal foliation (resp. Riemannian foliation). Conversely, the leaves of any conformal foliation (resp. Riemannian foliation) are locally the fibres of a horizontally conformal (resp. Riemannian) submersion, see [2].

Let (G, g) be a 4-dimensional Lie group equipped with a left-invariant Riemannian metric g and K be a 2-dimensional subgroup of G. Let \(\mathfrak {k}\) and \(\mathfrak {g}\) be the Lie algebras of K and G, respectively. Let \(\mathfrak {m}\) be the two dimensional orthogonal complement of \(\mathfrak {k}\) in \(\mathfrak {g}\) with respect to the Riemannian metric g on G. By \(\mathcal {V}\) we denote the integrable distribution generated by \(\mathfrak {k}\) and by \(\mathcal {H}\) its orthogonal distribution given by \(\mathfrak {m}\). Further let \(\mathcal {F}\) be the foliation of G tangent to \(\mathcal {V}\). Let the set

$$\begin{aligned} \mathcal {B}=\{X,Y,Z,W\} \end{aligned}$$

be an orthonormal basis for the Lie algebra \(\mathfrak {g}\) of G, such that Z, W generate the subalgebra \(\mathfrak {k}\) and \([W,Z]=\lambda \,W\) for some \(\lambda \in {\mathbb {R}}\). The elements \(X,Y\in \mathfrak {m}\) can clearly be chosen such that \(\mathcal {H}[X,Y]=r\,X\) for some \(r\in {\mathbb {R}}\).

If the 2-dimensional foliation \(\mathcal {F}\) is conformal with minimal leaves then, following [5], the Lie bracket relations of \(\mathfrak {g}\) take the following form

$$\begin{aligned}{}[W,Z]= & {} \lambda W, \nonumber \\ [Z,X]= & {} \alpha X +\beta Y+z_1 Z+w_1 W, \nonumber \\ [Z,Y]= & {} -\beta X+\alpha Y+z_2 Z+w_2 W, \nonumber \\ [W,X]= & {} a X +b Y+z_3 Z-z_1W, \nonumber \\ [W,Y]= & {} -b X +a Y+z_4 Z-z_2W, \nonumber \\ [Y,X]= & {} r X +\theta _1 Z+\theta _2 W, \end{aligned}$$
(2.1)

with real structure coefficients. To ensure that these equations actually define a Lie algebra they must satisfy the Jacobi equations. These are equivalent to a system of 14 second order homogeneous polynomial equations in 14 variables, see Appendix A. This has been solved in [5] and the solutions give a complete classifications of such Lie algebras. They form 20 multi-dimensional families which can be found therein.

For the foliation \(\mathcal {F}\) we state the following easy result describing the geometry of the situation.

Proposition 2.1

Let (G, g) be a 4-dimensional Lie group with Lie algebra \(\mathfrak {g}\) as above. Then

  1. (i)

    \(\mathcal {F}\) is totally geodesic if and only if \(z_1=z_2=z_3+w_1=z_4+w_2=0\),

  2. (ii)

    \(\mathcal {F}\) is Riemannian if and only if \(\alpha =a=0\), and

  3. (iii)

    \(\mathcal {H}\) is integrableif and only if \(\theta _1=\theta _2=0\).

3 The adapted almost hermitian structures

Let (G, g) be a 4-dimensional Riemannian Lie group as above. Further, let J be a left-invariant almost Hermitian structure on the tangent bundle TG adapted to the decomposition \(TG=\mathcal {V}\oplus \mathcal {H}\) i.e. a bundle isomorphism \(J:TG\rightarrow TG\) such that

$$\begin{aligned} J^2=-\text {id},\ \ J(\mathcal {V})=\mathcal {V}\ \ \text {and}\ \ J(\mathcal {H})=\mathcal {H}. \end{aligned}$$

The almost Hermitian structure J is left-invariant and compatible with the metric g. It can therefore be fully described by its action on an orthonormal basis

$$\begin{aligned} \mathcal {B}=\{X,Y,Z,Y\} \end{aligned}$$

of the Lie algebra \(\mathfrak {g}\) of G. Here the vector fields \(X,Y\in \mathcal {H}\) and \(Z,W\in \mathcal {V}\) can be chosen such that

$$\begin{aligned} J(X)=Y\ \ \text {and}\ \ J(Z)=W. \end{aligned}$$
(3.1)

Lemma 3.1

Let (G, g, J) be a 4-dimensional almost Hermitian Lie group as above. Then the structure J is almost Kähler \((\mathcal {A}\mathcal {K})\) if and only if

$$\begin{aligned} \theta _1=2\, a\ \ \text {and}\ \ \theta _2=-2\,\alpha . \end{aligned}$$

Proof

The Kähler form \(\omega\) satisfies \(\omega (E,F)=g(JE,F)\) for all \(E,F\in \mathfrak {g}\). It is well known that J is almost Kähler if and only if the corresponding Kähler form \(\omega\) is closed i.e. \(d\omega =0\).

$$\begin{aligned} d\omega (X,Y,Z)= & {} -g(J\left[ X,Y\right] ,Z) -g(J\left[ Y,Z\right] ,X) -g(J\left[ Z,X\right] ,Y)\\= & {} -g(J(-rX-\theta _1 Z-\theta _2 W),Z)\\&- g(J(\beta X-\alpha Y-z_2 Z-w_2 W),X)\\&-g(J(\alpha X +\beta Y+z_1 Z+w_1 W),Y)\\= & {} -g(-rY-\theta _1 W+\theta _2 Z,Z)\\&-g(\beta Y+\alpha X-z_2 W+w_2 Z,X)\\&-g(\alpha Y -\beta X+z_1 W-w_1 Z,Y)\\= & {} -\theta _2-2\,\alpha . \end{aligned}$$

The statement is then easily obtained by performing the corresponding computations in the other cases \(d\omega (W,X,Y)\), \(d\omega (Z,W,X)\) and \(d\omega (Y,Z,W)\). \(\square\)

Lemma 3.2

Let (G, g, J) be a 4-dimensional almost Hermitian Lie group as above. Then the almost complex structure J is integrable \((\mathcal {I})\) if and only if

$$\begin{aligned} 2z_1-z_4-w_2=0\ \ \text {and}\ \ 2z_2+z_3+w_1=0. \end{aligned}$$

Proof

It is well known that J is integrable if and only if the corresponding skew-symmetric Nijenhuis tensor N vanishes i.e. if

$$\begin{aligned} N(E,F) = [E,F] + J[JE,F] + J[E,JF] - [JE,JF]=0, \end{aligned}$$

for all \(E,F\in \mathfrak {g}\). It is obvious that \(N(X,Y)=N(Z,W)=0\). Using the fact that \(N(JE,JF)=-N(E,F)\) for all E, F it is clear that the only condition that has to be verified is \(N(X,Z) = 0\). Now

$$\begin{aligned} N(X,Z)= & {} [X,Z] + J[JX,Z] + J[X,JZ] - [JX,JZ]\\= & {} [X,Z] + J[Y,Z] + J[X,W] - [Y,W]\\= & {} -\alpha X -\beta Y-z_1 Z-w_1 W+J(\beta X-\alpha Y-z_2 Z-w_2 W)\\&-J( a X +b Y+z_3 Z-z_1W)-b X+a Y+z_4 Z-z_2W\\= & {} -\alpha X -\beta Y-z_1 Z-w_1 W+\beta Y+\alpha X-z_2 W+w_2 Z\\&- a Y +b X-z_3 W-z_1Z-b X+a Y+z_4 Z-z_2W\\= & {} -(2z_1-z_4-w_2)\,Z-(2z_2+z_3+w_1)\,W. \end{aligned}$$

\(\square\)

Corollary 3.3

Let (G, g, J) be a 4-dimensional almost Hermitian Lie group as above. Then the structure J is Kähler \((\mathcal {K})\) if and only if

$$\begin{aligned} 2z_1-z_4-w_2=0,\ \ 2z_2+z_3+w_1=0,\ \ \theta _1=2a\ \ \text {and}\ \ \theta _2=-2\,\alpha . \end{aligned}$$

Proof

The almost Hermitian structure J is Kähler if and only if it is almost Kähler and integrable. Hence the statement is an immediate consequence of Lemmas 3.1 and 3.2. \(\square\)

Remark 3.4

The 2-dimensional Lie algebra \(\mathfrak {k}\) of the subgroup K of G has orthonormal basis \(\{Z,W\}\), satisfying \([W,Z]=\lambda \,W\). This means that its simply connected universal covering group \(\widetilde{K}\) has constant Gaussian curvature \(\kappa =-\lambda ^2\). Thus it is either a hyperbolic disk \(H^2_\lambda\) or the flat Euclidean plane \({\mathbb {R}}^2\).

The 4-dimensional Riemannian Lie groups (G, g), with a left-invariant conformal foliation \(\mathcal {F}\) with minimal leaves, were classified in [5]. The purpose of this work is to in investigate their adapted almost Hermitian structure J, given by (3.1), in each case. Here we adopt the notation introduced in [5].

4 Case (A) - (\(\lambda \ne 0\) and \((\lambda -\alpha )^2+\beta ^2\ne 0\))

Example 4.1

(\(\mathfrak {g}_{1}(\lambda ,r,w_1,w_2)\)) This is a 4-dimensional family of solvable Lie algebras obtained by assuming that \(r\ne 0\), see [5]. The Lie bracket relations are given by

$$\begin{aligned}{}[W,Z]= & {} \lambda W,\\ [Z,X]= & {} w_1 W,\\ [Z,Y]= & {} w_2 W,\\ [Y,X]= & {} r X+\frac{rw_1}{\lambda }W. \end{aligned}$$

\((\mathcal {AK}):\) According to Lemma 3.1, the adapted almost Hermitian structure J is almost Kähler if and only if \(\theta _1=2a\) and \(\theta _2=-2\alpha\). We see from the bracket relations that \(a=\theta _1=\alpha =0\), hence \(\theta _2=0\). But \(\lambda \theta _2=rw_1\) which implies that \(rw_1=0\). Since \(r\ne 0\) we conclude that the family is in the almost Kähler class if and only if \(w_1=0\). This provides the 3-dimensional family \(\mathfrak {g}_1^{\mathcal {AK}}(\lambda ,r,w_2)\), given by

$$\begin{aligned}{}[W,Z]= & {} \lambda W,\\ [Z,Y]= & {} w_2 W,\\ [Y,X]= & {} r X. \end{aligned}$$

Here the corresponding simply connected Lie groups are semidirect products \(H^2_r\ltimes H^2_\lambda\) of hyperbolic disks, see Remark 3.4.

\((\mathcal {I}):\) According to Lemma 3.2 the structure J is integrable if and only if

$$\begin{aligned} 2z_2+z_3+w_1=0\ \ \text {and}\ \ 2z_1-z_4-w_2=0. \end{aligned}$$

Because \(z_1=z_2=z_3=z_4=0\) we conclude that \(w_1=w_2=0\). From this we yield the 2-dimensional family \(\mathfrak {g}_1^{\mathcal {I}}(\lambda ,r)\) satisfying

$$\begin{aligned}{}[W,Z]= & {} \lambda W,\\ [Y,X]= & {} r X. \end{aligned}$$

\((\mathcal {K}):\) The family \(\mathfrak {g}_1^{\mathcal {K}}(\lambda ,r)=\mathfrak {g}_1^{\mathcal {I}}(\lambda ,r)\) is contained in \(\mathfrak {g}_1^{\mathcal {AK}}(\lambda ,r,w_2)\) and thus Kähler, according to Corollary 3.3. Here the corresponding simply connected Lie groups are direct products \(H^2_r\times H^2_\lambda\) of hyperbolic disks.

Example 4.2

(\(\mathfrak {g}_{2}(\lambda ,\alpha ,\beta ,w_1,w_2)\)) Here we have the 5-dimensional family of solvable Lie algebras with the bracket relations

$$\begin{aligned}{}[W,Z]= & {} \lambda W,\\ [Z,X]= & {} \alpha X+\beta Y+w_1 W,\\ [Z,Y]= & {} -\beta X+\alpha Y+w_2 W. \end{aligned}$$

\((\mathcal {AK}):\) Since \(\theta _1=\theta _2=a=0\), the family \(\mathfrak {g}_2\) is almost Kähler if and only if \(\alpha =0.\) This yields a 4-dimensional family \(\mathfrak {g}_{2}^{\mathcal {AK}}(\lambda ,\beta ,w_1,w_2)\) with the Lie bracket relations

$$\begin{aligned}{}[W,Z]= & {} \lambda W,\\ [Z,X]= & {} \beta Y+w_1 W,\\ [Z,Y]= & {} -\beta X+w_2 W. \end{aligned}$$

\((\mathcal {I}):\) Because \(z_1=z_2=z_3=z_4=0\), the structure J is integrable if and only if \(w_1=w_2=0\). This provides us with a 3-dimensional family \(\mathfrak {g}_{2}^{\mathcal {I}}(\lambda ,\alpha ,\beta )\) given by

$$\begin{aligned}{}[W,Z]= & {} \lambda W,\\ [Z,X]= & {} \alpha X+\beta Y,\\ [Z,Y]= & {} -\beta X+\alpha Y. \end{aligned}$$

The corresponding simply connected Lie group is clearly a semidirect product \(H^2_\lambda \ltimes {\mathbb {R}}^2\).

\((\mathcal {K}):\) The family \(\mathfrak {g}_2\) is Kähler if and only if it is both almost Kähler and integrable i.e. if \(\alpha =w_1=w_2=0.\) Here we obtain the 2-dimensional family \(\mathfrak {g}_{2}^{\mathcal {K}}(\lambda ,\beta )\) given by

$$\begin{aligned}{}[W,Z]= & {} \lambda W,\\ [Z,X]= & {} \beta Y,\\ [Z,Y]= & {} -\beta X. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \(H^2_\lambda \ltimes {\mathbb {R}}^2\).

Example 4.3

(\(\mathfrak {g}_{3}(\alpha ,\beta ,w_1,w_2,\theta _2)\)) This is a 5-dimensional family of solvable Lie algebras with \(r=\theta _1=0,\) \(\theta _2\ne 0\) and \(\lambda =-2\alpha\). The bracket relations are given by

$$\begin{aligned}{}[W,Z]= & {} -2\alpha W,\\ [Z,X]= & {} \alpha X+\beta Y+w_1 W,\\ [Z,Y]= & {} -\beta X+\alpha Y+w_2 W,\\ [Y,X]= & {} \theta _2W. \end{aligned}$$

\((\mathcal {AK}):\) Here \(a=\theta _1=0,\) \(\lambda =-2\alpha \ne 0\) and \(\theta _2\ne 0.\) Thus, the family \(\mathfrak {g}_3\) is almost Kähler if and only if \(\theta _2=-2\alpha =\lambda \ne 0,\) which gives the 4-dimensional family \(\mathfrak {g}_{3}^{\mathcal {AK}}(\alpha ,\beta ,w_1,w_2)\) with

$$\begin{aligned}{}[W,Z]= & {} -2\alpha W,\\ [Z,X]= & {} \alpha X+\beta Y+w_1 W,\\ [Z,Y]= & {} -\beta X+\alpha Y+w_2 W,\\ [Y,X]= & {} -2\alpha W. \end{aligned}$$

\((\mathcal {I}):\) Because \(z_1=z_2=z_3=z_4=0\), we see that the structure J is integrable if and only if \(w_1=w_2=0,\) and obtain the family \(\mathfrak {g}_{3}^{\mathcal {I}}(\alpha ,\beta ,\theta _2)\) with

$$\begin{aligned}{}[W,Z]= & {} -2\alpha W,\\ [Z,X]= & {} \alpha X+\beta Y,\\ [Z,Y]= & {} -\beta X+\alpha Y,\\ [Y,X]= & {} \theta _2W. \end{aligned}$$

(\(\mathcal {K}):\) The family \(\mathfrak {g}_3\) is Kähler if and only if \(\theta _2=-2\alpha \ne 0\) and \(w_1=w_2=0.\) This provides \(\mathfrak {g}_{3}^\mathcal {K}(\alpha ,\beta )\) with

$$\begin{aligned}{}[W,Z]= & {} -2\alpha W,\\ [Z,X]= & {} \alpha X+\beta Y,\\ [Z,Y]= & {} -\beta X+\alpha Y,\\ [Y,X]= & {} -2\alpha W. \end{aligned}$$

5 Case (B) - (\(\lambda \ne 0\) and \((\lambda -\alpha )^2+\beta ^2=0\))

Example 5.1

(\(\mathfrak {g}_{4}(\lambda ,z_2,w_1,w_2)\)) Here we have the 4-dimensional family of solvable Lie algebras with the bracket relations

$$\begin{aligned}{}[W,Z]= & {} \lambda W,\\ [Z,X]= & {} \lambda X+w_1 W,\\ [Z,Y]= & {} \lambda Y+z_2Z+w_2 W,\\ [W,Y]= & {} -z_2W,\\ [Y,X]= & {} -z_2 X-\frac{z_2w_1}{\lambda }W. \end{aligned}$$

\((\mathcal {AK}):\) From \(\theta _1=2a=0\), \(\alpha =\lambda \ne 0\) and \(\lambda \theta _2=-z_2 w_1\), we see that \(\mathfrak {g}_4\) is almost Kähler if and only if \(\theta _2=-2\alpha =-2\lambda .\) This implies that \(2\lambda ^2=w_1z_2\ne 0\) and we yield the 3-dimensional \(\mathfrak {g}_{4}^{\mathcal {AK}}(\lambda ,z_2,w_2)\) with the following Lie bracket relations

$$\begin{aligned}{}[W,Z]= & {} \lambda W,\\ [Z,X]= & {} \lambda X+\frac{2\lambda ^2}{z_2} W,\\ [Z,Y]= & {} \lambda Y+z_2Z+w_2 W,\\ [W,Y]= & {} -z_2W,\\ [Y,X]= & {} -z_2 X-2\lambda W. \end{aligned}$$

\((\mathcal {I}):\) Because \(z_1=z_3=z_4=0\), the structure J is integrable if and only if \(2z_2+w_1=w_2=0\). This gives a 2-dimensional family \(\mathfrak {g}_{4}^{\mathcal {I}}(\lambda ,z_2)\) with bracket relations

$$\begin{aligned}{}[W,Z]= & {} \lambda W,\\ [Z,X]= & {} \lambda X-2z_2 W,\\ [Z,Y]= & {} \lambda Y+z_2Z,\\ [W,Y]= & {} -z_2W,\\ [Y,X]= & {} -z_2 X+\frac{2z_2^2}{\lambda }W. \end{aligned}$$

\((\mathcal {K}):\) Here the Kähler condition is never satisfied. This requires \(\lambda ^2+z_2^2=0\), which clearly has no real solutions for \(\lambda \ne 0\). Hence \(\mathfrak {g}_{4}^{\mathcal {K}}=\emptyset\).

6 Case (C) - (\(\lambda =0\), \(r\ne 0\) and \((a\beta -\alpha b)\ne 0\))

Example 6.1

(\(\mathfrak {g}_{5}(\alpha ,a,\beta ,b,r)\)) This is the 5-dimensional family of solvable Lie algebras satisfying the bracket relations

$$\begin{aligned}{}[Z,X]=\,\, & {} \alpha X +\beta Y -\frac{r(\beta b-\alpha a)}{2(\alpha b-a\beta )} Z -\frac{r(\alpha ^2-\beta ^2)}{2(\alpha b-a\beta )} W,\\ [Z,Y]=\,\, & {} -\beta X+\alpha Y -\frac{r(\alpha b+\beta a)}{2(\alpha b-a\beta )} Z +\frac{r\alpha \beta }{(\alpha b-a\beta )} W,\\ [W,X]=\,\, & {} a X +b Y -\frac{r(b^2-a^2)}{2(\alpha b-a\beta )} Z -\frac{r(\alpha a-\beta b)}{2(\alpha b-a\beta )}W,\\ [W,Y]=\,\, & {} -b X +a Y -\frac{rab }{(\alpha b-a\beta )} Z +\frac{r(\alpha b+\beta a)}{2(\alpha b-a\beta )}W,\\ [Y,X]=\,\, & {} r X +\frac{ar^2}{2(\alpha b-a\beta )} Z -\frac{\alpha r^2}{2(\alpha b-a\beta )} W. \end{aligned}$$

\((\mathcal {AK}):\) In this case we have the two relations

$$\begin{aligned} \theta _1=\frac{ar^2}{2(\alpha b-a\beta )} \ \ \text {and} \ \ \theta _2=-\frac{\alpha r^2}{2(\alpha b-a\beta )}. \end{aligned}$$

They imply that the structure J is almost Kähler if and only if

$$\begin{aligned} 2a=\frac{ar^2}{2(\alpha b-a\beta )} \ \ \text {and} \ \ 2\alpha =\frac{\alpha r^2}{2(\alpha b-a\beta )}. \end{aligned}$$

The condition \(a\beta -\alpha b\ne 0\) shows that not both a and \(\alpha\) can be zero, which implies that \(r^2=4{(\alpha b-a\beta )}>0\). If we substitute this into the above system, we then yield the following two solutions

$$\begin{aligned}{}[Z,X]= & {} \alpha X +\beta Y \mp \frac{(\beta b-\alpha a)}{\sqrt{\alpha b-a\beta }} Z \mp \frac{(\alpha ^2-\beta ^2)}{\sqrt{\alpha b-a\beta }} W,\\ [Z,Y]= & {} -\beta X+\alpha Y \mp \frac{(\alpha b+\beta a)}{\sqrt{\alpha b-a\beta }} Z \pm \frac{2\alpha \beta }{\sqrt{\alpha b-a\beta }} W,\\ [W,X]= & {} a X +b Y \mp \frac{(b^2-a^2)}{\sqrt{\alpha b-a\beta }} Z \mp \frac{(\alpha a-\beta b)}{\sqrt{\alpha b-a\beta }}W,\\ [W,Y]= & {} -b X +a Y \mp \frac{2ab }{\sqrt{\alpha b-a\beta }} Z \pm \frac{(\alpha b+\beta a)}{\sqrt{\alpha b-a\beta }}W,\\ [Y,X]= & {} \pm 2\sqrt{\alpha b-a\beta }\, X +2a\, Z -2\alpha \, W. \end{aligned}$$

Here we have obtained two 4-dimensional families \(\mathfrak {g}_{5}^{\mathcal {A}\mathcal {K}}(\alpha ,a,\beta ,b)^\pm\) satisfying the almost Kähler condition.

\((\mathcal {I}):\) Here we are assuming that \(r\ne 0\), \(a\beta -\alpha b\ne 0\) so we have

$$\begin{aligned} z_1= & {} -\frac{r(\beta b-\alpha a)}{2(\alpha b-a\beta )}, \ \ z_2=-\frac{r(\alpha b+\beta a)}{2(\alpha b-a\beta )}, \ \ z_3=-\frac{r(b^2-a^2)}{2(\alpha b-a\beta )}, \\ z_4= & {} -\frac{2rab }{2(\alpha b-a\beta )}, \ \ w_1=-\frac{r(\alpha ^2-\beta ^2)}{2(\alpha b-a\beta )},\ \ w_2=\frac{2r\alpha \beta }{2(\alpha b-a\beta )}. \end{aligned}$$

From this we observe that J is integrable if and only if

$$\begin{aligned} 2(\alpha b+\beta a) +b^2-a^2 +\alpha ^2-\beta ^2=0 \end{aligned}$$

and

$$\begin{aligned} \beta b-\alpha a -ab +\alpha \beta =0. \end{aligned}$$

A simple calculation shows that J is integrable if and only if \(a=\beta\) and \(b=-\alpha\). This provides us with the 3-dimensional family \(\mathfrak {g}_{5}^\mathcal {I}(\alpha ,\beta ,r)\) fulfilling the bracket relations

$$\begin{aligned}{}[Z,X]= & {} \alpha X +\beta Y -\frac{r\alpha \beta }{(\beta ^2+\alpha ^2)} Z+\frac{r(\alpha ^2-\beta ^2)}{2(\beta ^2+\alpha ^2)} W,\\ [Z,Y]= & {} -\beta X+\alpha Y -\frac{r(\alpha ^2-\beta ^2)}{2(\beta ^2+\alpha ^2)} Z-\frac{r\alpha \beta }{(\beta ^2+\alpha ^2)} W,\\ [W,X]= & {} \beta X -\alpha Y +\frac{r(\alpha ^2-\beta ^2)}{2(\beta ^2+\alpha ^2)} Z+\frac{r\alpha \beta }{(\beta ^2+\alpha ^2)}W,\\ [W,Y]= & {} \alpha X +\beta Y -\frac{r\alpha \beta }{(\beta ^2+\alpha ^2)} Z+\frac{r(\alpha ^2-\beta ^2)}{2(\beta ^2+\alpha ^2)}W,\\ [Y,X]= & {} r X -\frac{\beta r^2}{2(\beta ^2+\alpha ^2)} Z+\frac{\alpha r^2}{2(\beta ^2+\alpha ^2)} W. \end{aligned}$$

\((\mathcal {K}):\) Combining the observations \(r^2=4{(\alpha b-a\beta )}>0\) from \((\mathcal {A}\mathcal {K})\) and \(a=\beta\), \(b=-\alpha\) from \((\mathcal {I})\) we see that the Kähler condition is never satisfied for the Lie algebra \(\mathfrak {g}_5\), so \(\mathfrak {g}_{5}^{\mathcal {K}}=\emptyset\).

7 Case (D) - (\(\lambda =0\), \(r\ne 0\) and \((a\beta -\alpha b)=0\))

Example 7.1

(\(\mathfrak {g}_{6}(z_1,z_2,z_3,r,\theta _1,\theta _2)\)) Here we have a 6-dimensional family of solvable Lie algebras with \(z_1^2=-w_1z_3\ne 0\),

$$\begin{aligned} z_4 =\frac{z_3(r+2z_2)}{2z_1},\ \ w_1 = -\frac{z_1^2}{z_3} \ \ \text {and}\ \ w_2 = \frac{z_1(r-2z_2)}{2z_3}. \end{aligned}$$

In this case the Lie bracket relations are given by

$$\begin{aligned}{}[Z,X]=\,\, & {} z_1 Z-\frac{z_1^2}{z_3} W,\\ [Z,Y]=\,\, & {} z_2 Z+\frac{z_1(r-2z_2)}{2z_3} W,\\ [W,X]=\,\, & {} z_3 Z-z_1W,\\ [W,Y]=\,\, & {} \frac{z_3(r+2z_2)}{2z_1} Z-z_2W,\\ [Y,X]=\,\, & {} r X+\theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {AK}):\) Because \(a=\alpha =0\), the structure J is almost Kähler if and only if \(\theta _1=\theta _2=0\). This provides the 4-dimensional family \(\mathfrak {g}_{6}^{\mathcal {AK}}(z_1,z_2,z_3,r)\) with the bracket relations

$$\begin{aligned}{}[Z,X]=\,\, & {} z_1 Z-\frac{z_1^2}{z_3} W,\\ [Z,Y]=\,\, & {} z_2 Z+\frac{z_1(r-2z_2)}{2z_3} W,\\ [W,X]=\,\, & {} z_3 Z-z_1W,\\ [W,Y]=\,\, & {} \frac{z_3(r+2z_2)}{2z_1} Z-z_2W,\\ [Y,X]=\,\, & {} r X. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \(H^2_r\ltimes {\mathbb {R}}^2\).

\((\mathcal {I}):\) We note that

$$\begin{aligned} z_1\ne 0, \ \ z_4=\frac{z_3(r+2z_2)}{2z_1}, \ \ w_1=-\frac{z_1^2}{z_3} \ \ \text {and} \ \ w_2=\frac{z_1(r-2z_2)}{2z_3}. \end{aligned}$$

Here the structure J is integrable if and only if

$$\begin{aligned} z_1^2=(2z_2+z_3)z_3 \ \ \text {and} \ \ z_1^2(4z_3-r+2z_2)=z_3^2(r+2z_2). \end{aligned}$$

By multiplying the first equation by \((4z_3-r+2z_2)\), we see that these two conditions imply

$$\begin{aligned} (2z_2+z_3)(4z_3-r+2z_2)=z_3(r+2z_2), \end{aligned}$$

which can be rewritten as

$$\begin{aligned} (z_2+z_3)(2(z_2+z_3)-r)=0. \end{aligned}$$

If \(z_2+z_3=0\), our earlier conditions give \(z_1^2+z_3^2=0\), which is not possible for nonzero real constants. Thus we must have \(z_2+z_3\ne 0\). This leaves us with \(r=2(z_2+z_3)\). We conclude that J is integrable if and only if

$$\begin{aligned} r=\frac{z_1^2+z_3^2}{z_3} \ \ \text {and} \ \ z_2=\frac{z_1^2-z_3^2}{2z_3}. \end{aligned}$$

This leads to the following bracket relations for the 4-dimensional family \(\mathfrak {g}_{6}^\mathcal {I}(z_1,z_3,\theta _1,\theta _2)\)

$$\begin{aligned}{}[Z,X]= & {} z_1 Z-\frac{z_1^2}{z_3} W,\\ [Z,Y]= & {} \frac{z_1^2-z_3^2}{2z_3} Z+z_1 W,\\ [W,X]= & {} z_3 Z-z_1W,\\ [W,Y]= & {} z_1 Z-\frac{z_1^2-z_3^2}{2z_3}W,\\ [Y,X]= & {} \frac{z_1^2+z_3^2}{z_3} X+\theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {K}):\) In this case the Kähler condition is fulfilled if and only if

$$\begin{aligned} \theta _1=\theta _2=0, \ \ r=\frac{z_1^2+z_3^2}{z_3} \ \ \text {and} \ \ z_2=\frac{z_1^2-z_3^2}{2z_3}. \end{aligned}$$

Hence we have the 2-dimensional family \(\mathfrak {g}_{6}^\mathcal {K}(z_1,z_3)\) with bracket relations

$$\begin{aligned}{}[Z,X]=\,\, & {} z_1 Z-\frac{z_1^2}{z_3} W,\\ [Z,Y]=\,\, & {} \frac{z_1^2-z_3^2}{2z_3} Z+z_1 W,\\ [W,X]=\,\, & {} z_3 Z-z_1W,\\ [W,Y]=\,\, & {} z_1 Z-\frac{z_1^2-z_3^2}{2z_3}W,\\ [Y,X]=\,\, & {} \frac{z_1^2+z_3^2}{z_3} X. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \(H^2_{r}\ltimes {\mathbb {R}}^2\).

Example 7.2

(\(\mathfrak {g}_{7}(z_2,w_1,w_2,\theta _1,\theta _2)\)) This is a 5-dimensional family of solvable Lie algebras with \(z_1=z_3=z_4=0\), \(r=2z_2\) and \(w_1\ne 0\). The bracket relations are

$$\begin{aligned}{}[Z,X]=\,\, & {} w_1 W,\\ [Z,Y]=\,\, & {} z_2 Z+w_2 W,\\ [W,Y]=\,\, & {} -z_2 W,\\ [Y,X]=\,\, & {} 2z_2 X+\theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {AK}):\) Because \(\alpha =a=0\), the family belongs to the almost Kähler class if and only if \(\theta _1=\theta _2=0\). This gives the 3-dimensional family \(\mathfrak {g}_{7}^{\mathcal {AK}}(z_2,w_1,w_2)\) with

$$\begin{aligned}{}[Z,X]=\,\, & {} w_1 W,\\ [Z,Y]=\,\, & {} z_2 Z+w_2 W,\\ [W,Y]=\,\, & {} -z_2 W,\\ [Y,X]=\,\, & {} 2z_2 X. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \(H^2_{2z_2}\ltimes {\mathbb {R}}^2\).

\((\mathcal {I}):\) We have \(z_1=z_3=z_4=0.\) Thus the structure J is integrable if and only if \(2z_2+w_1=w_2=0\) and we yield a 3-dimensional family \(\mathfrak {g}_{7}^{\mathcal {I}}(z_2,\theta _1,\theta _2)\) given by

$$\begin{aligned}{}[Z,X]=\,\, & {} -2z_2 W,\\ [Z,Y]=\,\, & {} z_2 Z,\\ [W,Y]=\,\, & {} -z_2 W,\\ [Y,X]=\,\, & {} 2z_2 X+\theta _1 Z+\theta _2 W. \end{aligned}$$

Note that \(z_2\ne 0\).

\((\mathcal {K}):\) The family \(\mathfrak {g}_7\) is of the Kähler class if and only if \(\theta _1=\theta _2=0\), \(w_2=0\) and \(w_1=-2z_2\ne 0\) providing the 1-dimensional family \(\mathfrak {g}_{7}^{\mathcal {K}}(z_2)\) with

$$\begin{aligned}{}[Z,X]=\,\, & {} -2z_2 W,\\ [Z,Y]=\,\, & {} z_2 Z,\\ [W,Y]=\,\, & {} -z_2 W,\\ [Y,X]=\,\, & {} 2z_2 X. \end{aligned}$$

The corresponding simply connected Lie group is clearly a semidirect product \(H^2_{2z_2}\ltimes {\mathbb {R}}^2\).

Example 7.3

(\(\mathfrak {g}_{8}(z_2,z_4,w_2,r,\theta _1,\theta _2)\)) Here we have the 6-dimensional family of solvable Lie algebras with bracket relations

$$\begin{aligned}{}[Z,Y]=\,\, & {} z_2 Z+w_2 W,\\ [W,Y]=\,\, & {} z_4 Z-z_2 W,\\ [Y,X]=\,\, & {} r X+\theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {AK}):\) Since \(\alpha =a=0\), the family is almost Kähler if and only if \(\theta _1=\theta _2=0,\) giving the 4-dimensional family \(\mathfrak {g}_{8}^{\mathcal {AK}}(z_2,z_4,w_2,r)\) with

$$\begin{aligned}{}[Z,Y]=\,\, & {} z_2 Z+w_2 W,\\ [W,Y]=\,\, & {} z_4 Z-z_2 W,\\ [Y,X]=\,\, & {} r X. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \(H^2_r\ltimes {\mathbb {R}}^2\).

\((\mathcal {I}):\) We have \(z_1=z_3=w_1=0,\) so J is integrable if and only if \(z_2=z_4+w_2=0\), so we yield the family \(\mathfrak {g}_{8}^{\mathcal {I}}(w_2,r,\theta _1,\theta _2)\) with

$$\begin{aligned}{}[Z,Y]=\,\, & {} w_2 W,\\ [W,Y]=\,\, & {} -w_2 Z,\\ [Y,X]=\,\, & {} r X+\theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {K}):\) The family is Kähler if and only if \(\theta _1=\theta _2=z_2=z_4+w_2=0.\) We get the 2-dimensional family \(\mathfrak {g}_{8}^{\mathcal {K}}(w_2,r)\) with

$$\begin{aligned}{}[Z,Y]=\,\, & {} w_2 W,\\ [W,Y]=\,\, & {} -w_2 Z,\\ [Y,X]=\,\, & {} r X. \end{aligned}$$

The corresponding simply connected Lie group is clearly a semidirect product \(H^2_{r}\ltimes {\mathbb {R}}^2\)

Example 7.4

(\(\mathfrak {g}_{9}(z_2,z_3,z_4,\theta _1,\theta _2)\)) This is the 5-dimensional family of solvable Lie algebras with \(z_3\ne 0\) and bracket relations

$$\begin{aligned}{}[Z,Y]=\,\, & {} z_2 Z,\\ [W,X]=\,\, & {} z_3 Z,\\ [W,Y]=\,\, & {} z_4 Z-z_2 W,\\ [Y,X]=\,\, & {} -2z_2 X+\theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {AK}):\) Because \(\alpha =a=0\), we see that the family is in the almost Kähler class if and only if \(\theta _1=\theta _2=0,\) providing \(\mathfrak {g}_{9}^{\mathcal {AK}}(z_2,z_3,z_4)\) with

$$\begin{aligned}{}[Z,Y]=\,\, & {} z_2 Z,\\ [W,X]=\,\, & {} z_3 Z,\\ [W,Y]=\,\, & {} z_4 Z-z_2 W,\\ [Y,X]=\,\, & {} -2z_2 X. \end{aligned}$$

\((\mathcal {I}):\) From \(z_1=w_1=w_2=0,\) we find that J is integrable if and only if \(2z_2+z_3=z_4=0,\) from which we yield \(\mathfrak {g}_{9}^{\mathcal {I}}(z_2,\theta _1,\theta _2)\) with the relations

$$\begin{aligned}{}[Z,Y]=\,\, & {} z_2 Z,\\ [W,X]=\,\, & {} -2z_2 Z,\\ [W,Y]=\,\, & {} -z_2 W,\\ [Y,X]=\,\, & {} -2z_2 X+\theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {K}):\) The family is Kähler if and only if \(\theta _1=\theta _2=0,\) \(z_3=-2z_2\) and \(z_4=0.\) Here we obtain the 1-dimensional family \(\mathfrak {g}_{9}^{\mathcal {K}}(z_2)\) with

$$\begin{aligned}{}[Z,Y]=\,\, & {} z_2 Z,\\ [W,X]=\,\, & {} -2z_2 Z,\\ [W,Y]=\,\, & {} -z_2 W,\\ [Y,X]=\,\, & {} -2z_2 X. \end{aligned}$$

The corresponding simply connected Lie group is clearly a semidirect product \(H^2_{2z_2}\ltimes {\mathbb {R}}^2\)

8 Case (E) - (\(\lambda =0\), \(r=0\) and \(\alpha b-a\beta \ne 0\))

Example 8.1

(\(\mathfrak {g}_{10}(\alpha ,a,\beta ,b)\)) Here we have the 4-dimensional family of solvable Lie algebras with the bracket relations

$$\begin{aligned}{}[Z,X]=\,\, & {} \alpha X+ \beta Y,\\ [Z,Y]=\,\, & {} -\beta X+\alpha Y,\\ [W,X]=\,\, & {} a X +b Y,\\ [W,Y]=\,\, & {} -b X +a Y. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

\((\mathcal {AK}):\) Since \(\theta _1=\theta _2=0\), we see that the family is almost Kähler if and only if \(a=\alpha =0.\) But then the requirement \(\alpha b-a\beta \ne 0\) is not true, so this family can not be almost Kähler, so \(\mathfrak {g}_{10}^{\mathcal {A}\mathcal {K}}=\{0\}\).

\((\mathcal {I}):\) Because \(z_1=z_2=z_3=z_4=w_1=w_2=0\), we can see that the structure J is always integrable in this family, therefore \(\mathfrak {g}_{10}^{\mathcal {I}}=\mathfrak {g}_{10}(\alpha ,a,\beta ,b)\).

\((\mathcal {K}):\) The Lie algebra \(\mathfrak {g}_{10}\) is never in the Kähler class, hence \(\mathfrak {g}_{10}^{\mathcal {K}}=\emptyset\).

9 Case (F) - (\(\lambda =0\), \(r=0\) and \(\alpha b-a\beta = 0\))

In this case our analysis divides into disjoint cases parametrized by \(\Lambda =(\alpha ,a,\beta ,b)\). The variables are assumed to be zero if and only if they are marked by 0. For example, if \(\Lambda =(0,a,\beta ,0)\) then the two variables \(\alpha\) and b are assumed to be zero and a and \(\beta\) to be non-zero.

Example 9.1

(\(\mathfrak {g}_{11}(z_1,z_2,z_3,w_1,\theta _1,\theta _2)\)) This is a 6-dimensional family of solvable Lie algebras. Here \(\Lambda =(0,0,0,0)\) and \(z_1\ne 0\), giving the Lie bracket relations

$$\begin{aligned}{}[Z,X]=\,\, & {} z_1 Z+w_1 W,\\ [Z,Y]=\,\, & {} z_2 Z+\frac{z_2w_1}{z_1} W,\\ [W,X]=\,\, & {} z_3 Z-z_1W,\\ [W,Y]=\,\, & {} \frac{z_2z_3}{z_1} Z-z_2 W,\\ [Y,X]=\,\, & {} \theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {AK}):\) Because \(\alpha =a=0\), we see that \(\mathfrak {g}_{11}\) is in the almost Kähler class if and only if \(\theta _1=\theta _2=0.\) This gives a 4-dimensional family \(\mathfrak {g}_{11}^{\mathcal {AK}}(z_1,z_2,z_3,w_1)\) with

$$\begin{aligned}{}[Z,X]=\,\, & {} z_1 Z+w_1 W,\\ [Z,Y]=\,\, & {} z_2 Z+\frac{z_2w_1}{z_1} W,\\ [W,X]=\,\, & {} z_3 Z-z_1W,\\ [W,Y]=\,\, & {} \frac{z_2z_3}{z_1} Z-z_2 W. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

\((\mathcal {I}):\) From the relations

$$\begin{aligned} z_1\ne 0, \ \ z_4=\frac{z_2 z_3}{z_1} \ \ \text {and} \ \ w_2=\frac{z_2 w_1}{z_1}, \end{aligned}$$

we see that the structure J is integrable if and only if

$$\begin{aligned} z_3+w_1=-2z_2\ \ \text {and} \ \ 2z_1^2-z_2(z_3+w_1)=0. \end{aligned}$$

This implies the impossible \(z_1^2+z_2^2=0\), so the structure J is never integrable, therefore \(\mathfrak {g}_{11}^{\mathcal {K}}=\{0\}\).

\((\mathcal {K}):\) This family \(\mathfrak {g}_{11}\) is never in the Kähler class, so \(\mathfrak {g}_{11}^{\mathcal {K}}=\emptyset\).

Example 9.2

(\(\mathfrak {g}_{12}(z_3,w_1,w_2,\theta _1,\theta _2)\)) Here we have a 5-dimensional family of solvable Lie algebras. Here \(\Lambda =(0,0,0,0)\), \(z_1=0\) and \(w_1\ne 0\), which gives

$$\begin{aligned} z_2=0\ \ \text {and}\ \ z_4=\frac{z_3w_2}{w_1} \end{aligned}$$

and the Lie bracket relations

$$\begin{aligned}{}[Z,X]=\,\, & {} w_1 W,\\ [Z,Y]=\,\, & {} w_2 W,\\ [W,X]=\,\, & {} z_3 Z,\\ [W,Y]=\,\, & {} \frac{z_3w_2}{w_1} Z,\\ [Y,X]=\,\, & {} \theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {AK}):\) Since \(\alpha =a=0\), this family is almost Kähler if and only if \(\theta _1=\theta _2=0.\) This yields the 3-dimensional family \(\mathfrak {g}^{\mathcal {AK}}_{12}(z_3,w_1,w_2)\) with the relations

$$\begin{aligned}{}[Z,X]=\,\, & {} w_1 W,\\ [Z,Y]=\,\, & {} w_2 W,\\ [W,X]=\,\, & {} z_3 Z,\\ [W,Y]=\,\, & {} \frac{z_3w_2}{w_1} Z. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

\((\mathcal {I}):\) For \(\mathfrak {g}_{12}\) we have

$$\begin{aligned} z_1=z_2=0, \ \ z_4=\frac{z_3w_2}{w_1} \ \ \text {and} \ \ w_1\ne 0. \end{aligned}$$

We see that the structure J is integrable if and only if

$$\begin{aligned} z_3+w_1=\frac{z_3w_2}{w_1}+w_2=0, \end{aligned}$$

or equivalently, \(z_3=-w_1\ne 0\). Here we yield the 4-dimensional family \(\mathfrak {g}_{12}^{\mathcal {I}}(w_1,w_2,\theta _1,\theta _2)\) which has the Lie bracket relations

$$\begin{aligned}{}[Z,X]=\,\, & {} w_1 W,\\ [Z,Y]=\,\, & {} w_2 W,\\ [W,X]=\,\, & {} -w_1 Z,\\ [W,Y]=\,\, & {} -w_2 Z,\\ [Y,X]=\,\, & {} \theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {K}):\) The family is Kähler if and only if \(\theta _1=\theta _2=0\) and \(z_3=-w_1\ne 0,\) which gives the 2-dimensional family \(\mathfrak {g}_{12}^{\mathcal {K}}(w_1,w_2)\) with relations

$$\begin{aligned}{}[Z,X]=\,\, & {} w_1 W,\\ [Z,Y]=\,\, & {} w_2 W,\\ [W,X]=\,\, & {} -w_1 Z,\\ [W,Y]=\,\, & {} -w_2 Z. \end{aligned}$$

The corresponding simply connected Lie group is clearly a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

Example 9.3

(\(\mathfrak {g}_{13}(z_3,z_4,\theta _1,\theta _2)\)) This is a 4-dimensional family of nilpotent Lie algebras. If \(\theta _2 = 0\) the Lie algebra is 2-step nilpotent, whereas it is 3-step nilpotent otherwise. Here \(\Lambda =(0,0,0,0)\), \(z_1=w_1=0\) and \(z_3\ne 0\), which gives \(z_2=w_2=0\) and the solutions

$$\begin{aligned}{}[W,X]=\,\, & {} z_3 Z,\\ [W,Y]=\,\, & {} z_4 Z,\\ [Y,X]=\,\, & {} \theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {AK}):\) Because \(\alpha =a=0\), we see that \(\mathfrak {g}_{13}\) family is in the almost Kähler class if and only if \(\theta _1=\theta _2=0\). We get the 2-dimensional family \(\mathfrak {g}_{13}^{\mathcal {AK}}(z_3,z_4)\) with

$$\begin{aligned}{}[W,X]=\,\, & {} z_3 Z,\\ [W,Y]=\,\, & {} z_4 Z. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

\((\mathcal {I}):\) The condition \(z_1=z_2=w_1=w_2=0\) gives that J is integrable if and only if \(z_3=z_4=0\), contradicting \(z_3\ne 0\). Hence \(\mathfrak {g}_{13}^{\mathcal {I}}=\emptyset\).

\((\mathcal {K}):\) This family \(\mathfrak {g}_{13}\) is never in the Kähler class, so \(\mathfrak {g}_{13}^{\mathcal {K}}=\emptyset\).

Example 9.4

(\(\mathfrak {g}_{14}(z_2,z_4,w_2,\theta _1,\theta _2)\)) Here we have the 5-dimensional family of solvable Lie algebras with \(\Lambda =(0,0,0,0)\) and \(z_1=z_3=w_1=0\) and the bracket relations

$$\begin{aligned}{}[Z,Y]=\,\, & {} z_2 Z+w_2 W,\\ [W,Y]=\,\, & {} z_4 Z-z_2 W,\\ [Y,X]=\,\, & {} \theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {AK}):\) The condition \(\alpha =a=0\) implies that this family is almost Kähler if and only if \(\theta _1=\theta _2=0.\) We get the 3-dimensional family \(\mathfrak {g}_{14}^{\mathcal {AK}}(z_2,z_4,w_2)\) given by

$$\begin{aligned}{}[Z,Y]=\,\, & {} z_2 Z+w_2 W,\\ [W,Y]=\,\, & {} z_4 Z-z_2 W. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

\((\mathcal {I}):\) From \(z_1=z_3=w_1=0\) we see that J is integrable if and only if \(z_2=0\) and \(z_4+w_2=0.\) This gives a 3-dimensional family \(\mathfrak {g}_{14}^{\mathcal {I}}(w_2,\theta _1,\theta _2)\) with Lie bracket relations

$$\begin{aligned}{}[Z,Y]=\,\, & {} w_2 W,\\ [W,Y]=\,\, & {} -w_2 Z,\\ [Y,X]=\,\, & {} \theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {K}):\) The family is Kähler if and only if \(\theta _1=\theta _2=0,\) \(z_2=0\) and \(z_4+w_2=0.\) We get a 1-dimensional family \(\mathfrak {g}_{14}^{\mathcal {K}}(w_2)\) with

$$\begin{aligned}{}[Z,Y]=\,\, & {} w_2 W,\\ [W,Y]=\,\, & {} -w_2 Z. \end{aligned}$$

The corresponding simply connected Lie group is clearly a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

Example 9.5

(\(\mathfrak {g}_{15}(\alpha ,w_1,w_2)\)) This is a 3-dimensional family of solvable Lie algebras with \(\Lambda =(\alpha ,0,0,0)\), which gives

$$\begin{aligned} z_1=z_2=z_3=z_4=\theta _1=\theta _2=0 \end{aligned}$$

and

$$\begin{aligned}{}[Z,X]=\,\, & {} \alpha X+w_1 W,\\ [Z,Y]=\,\, & {} \alpha Y+w_2 W. \end{aligned}$$

\((\mathcal {AK}):\) Here, \(a=\theta _1=\theta _2=0\) and \(\alpha \ne 0,\) so we see that the almost Kähler is never fulfilled, hence \(\mathfrak {g}_{15}^{\mathcal {AK}}=\emptyset\).

\((\mathcal {I}):\) The conditions \(z_1=z_2=z_3=z_4=0\) imply that J is integrable if and only if \(w_1=w_2=0.\) We yield a 1-dimensional family \(\mathfrak {g}_{15}^{\mathcal {I}}(\alpha )\) with

$$\begin{aligned}{}[Z,X]=\,\, & {} \alpha X,\\ [Z,Y]=\,\, & {} \alpha Y. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

\((\mathcal {K}):\) Here the Kähler condition is never satisfied, so \(\mathfrak {g}_{15}^{\mathcal {K}}=\emptyset\).

Example 9.6

(\(\mathfrak {g}_{16}(\beta ,w_1,w_2,\theta _1,\theta _2)\)) Here we have a 5-dimensional family of Lie algebras. They are not solvable in general. Here \(\Lambda =(0,0,\beta ,0)\), which implies \(z_1=z_2=z_3=z_4=0\). This gives

$$\begin{aligned}{}[Z,X]=\,\, & {} \beta Y+w_1 W,\\ [Z,Y]= & {} -\beta X+w_2 W,\\ [Y,X]=\,\, & {} \theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {AK}):\) We have \(\alpha =a=0,\) so this family is almost Kähler if and only if \(\theta _1=\theta _2=0.\) We get the 3-dimensional family \(\mathfrak {g}_{16}^{\mathcal {AK}}(\beta ,w_1,w_2)\) with

$$\begin{aligned}{}[Z,X]=\,\, & {} \beta Y+w_1 W,\\ [Z,Y]= & {} -\beta X+w_2 W. \end{aligned}$$

\((\mathcal {I}):\) We have \(z_1=z_2=z_3=z_4=0,\) so J is integrable if and only if \(w_1=w_2=0,\) giving the 3-dimensional family \(\mathfrak {g}_{16}^{\mathcal {I}}(\beta ,\theta _1,\theta _2)\) with

$$\begin{aligned}{}[Z,X]=\,\, & {} \beta Y,\\ [Z,Y]=\,\, & {} -\beta X,\\ [Y,X]=\,\, & {} \theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {K}):\) The family \(\mathfrak {g}_{16}\) is Kähler if and only if \(\theta _1=\theta _2=0\) and \(w_1=w_2=0.\) We obtain the 1-dimensional family \(\mathfrak {g}_{16}^{\mathcal {K}}(\beta )\) with Lie bracket relations

$$\begin{aligned}{}[Z,X]=\,\, & {} \beta Y,\\ [Z,Y]=\,\, & {} -\beta X. \end{aligned}$$

The corresponding simply connected Lie group is clearly a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

Example 9.7

(\(\mathfrak {g}_{17}(\alpha ,a,w_1,w_2)\)) This is a 4-dimensional family of solvable Lie algebras \(\Lambda =(\alpha ,a,0,0)\), which gives

$$\begin{aligned} z_1=-\frac{aw_1}{\alpha },\ \ z_2=-\frac{aw_2}{\alpha },\ \ z_3=-\frac{a^2w_1}{\alpha ^2},\ \ z_4=-\frac{a^2w_2}{\alpha ^2},\ \ \theta _1=0,\ \ \theta _2=0. \end{aligned}$$

Thus the bracker relations are

$$\begin{aligned}{}[Z,X]=\,\, & {} \alpha X-\frac{aw_1}{\alpha }Z+w_1 W,\\ [Z,Y]=\,\, & {} \alpha Y-\frac{aw_2}{\alpha }Z+w_2 W,\\ [W,X]=\,\, & {} a X-\frac{a^2w_1}{\alpha ^2} Z+\frac{aw_1}{\alpha }W,\\ [W,Y]=\,\, & {} a Y-\frac{a^2w_2}{\alpha ^2} Z+\frac{aw_2}{\alpha }W. \end{aligned}$$

\((\mathcal {AK}):\) Because \(\theta _1=\theta _2=0\), \(\alpha \ne 0\) and \(a\ne 0\), the almost Kähler condition is never satisfied, hence \(\mathfrak {g}_{17}^{\mathcal {AK}}=\{0\}\).

\((\mathcal {I}):\) From

$$\begin{aligned} z_1=-\frac{aw_1}{\alpha }, \ \ z_2=-\frac{aw_2}{\alpha }, \ \ z_3=-\frac{a^2w_1}{\alpha ^2} \ \ \text {and} \ \ z_4=-\frac{a^2w_2}{\alpha ^2} \end{aligned}$$

we see that the structure J is integrable if and only if

$$\begin{aligned} \left( \alpha ^2- {a^2}\right) w_1=2a\alpha w_2 \ \ \text {and} \ \ \left( \alpha ^2- {a^2}\right) w_2=-2a\alpha w_1. \end{aligned}$$
(9.1)

Since \(2a\alpha \ne 0\), we can divide and get

$$\begin{aligned} w_2=\frac{\alpha ^2-a^2}{2a\alpha }w_1 =-\frac{(\alpha ^2-a^2)^2}{4a^2\alpha ^2}w_2. \end{aligned}$$

If \(w_2\ne 0\), then we yield the impossible \((\alpha ^2+a^2)^2=0\). Thus J is integrable if and only if \(w_1=w_2=0\). This gives the 2-dimensional family \(\mathfrak {g}_{17}^{\mathcal {H}}(\alpha ,a)\) with

$$\begin{aligned}{}[Z,X]=\,\, & {} \alpha X,\\ [Z,Y]=\,\, & {} \alpha Y,\\ [W,X]=\,\, & {} a X,\\ [W,Y]=\,\, & {} a Y. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

\((\mathcal {K}):\) The family \(\mathfrak {g}_{17}\) never satisfies the Kähler condition, so \(\mathfrak {g}_{17}^{\mathcal {AK}}=\emptyset\).

Example 9.8

(\(\mathfrak {g}_{18}(\beta ,b,z_3,z_4,\theta _1,\theta _2)\)) Here we have a 6-dimensional family of Lie algebras. They are not solvable in general. Here \(\Lambda =(0,0,\beta ,b)\), which gives

$$\begin{aligned} z_1=\frac{\beta z_3}{b},\ \ z_2=\frac{\beta z_4}{b},\ \ w_1=-\frac{\beta ^2z_3}{b^2}\ \ \text {and}\ \ w_2=-\frac{\beta ^2z_4}{b^2}. \end{aligned}$$

The bracket relations satisfy

$$\begin{aligned}{}[Z,X]=\,\, & {} \beta Y+\frac{\beta z_3}{b} Z-\frac{\beta ^2z_3}{b^2} W,\\ [Z,Y]=\,\, & {} -\beta X+\frac{\beta z_4}{b} Z-\frac{\beta ^2z_4}{b^2} W,\\ [W,X]=\,\, & {} b Y+z_3 Z-\frac{\beta z_3}{b} W,\\ [W,Y]=\,\, & {} -b X+z_4 Z-\frac{\beta z_4}{b} W,\\ [Y,X]=\,\, & {} \theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {AK}):\) Since \(a=\alpha =0\), this family is almost Kähler if and only if \(\theta _1=\theta _2=0.\) We get the 4-dimensional family \(\mathfrak {g}_{18}^\mathcal {AK}(\beta ,b,z_3,z_4)\) with

$$\begin{aligned}{}[Z,X]=\,\, & {} \beta Y+\frac{\beta z_3}{b} Z-\frac{\beta ^2z_3}{b^2} W,\\ [Z,Y]=\,\, & {} -\beta X+\frac{\beta z_4}{b} Z-\frac{\beta ^2z_4}{b^2} W,\\ [W,X]=\,\, & {} b Y+z_3 Z-\frac{\beta z_3}{b} W,\\ [W,Y]=\,\, & {} -b X+z_4 Z-\frac{\beta z_4}{b} W. \end{aligned}$$

\((\mathcal {I}):\) For the family \(\mathfrak {g}_{18}\) we have

$$\begin{aligned} z_1=\frac{\beta z_3}{b}, \ \ z_2=\frac{\beta z_4}{b}, \ \ w_1=-\frac{\beta ^2z_3}{b^2} \ \ \text {and} \ \ w_2=-\frac{\beta ^2z_4}{b^2} \end{aligned}$$

and we can easily see that J is integrable if and only if

$$\begin{aligned} (\beta ^2-b^2)z_3=2\beta bz_4 \ \ \text {and} \ \ (\beta ^2-b^2)z_4=-2\beta bz_3. \end{aligned}$$

We notice that this system has the same form as that of (9.1). Thus the structure J is integrable if and only if \(z_3=z_4=0.\) Here we yield the 4-dimensional family \(\mathfrak {g}_{18}^\mathcal {I}(\beta ,b,\theta _1,\theta _2)\), where

$$\begin{aligned}{}[Z,X]=\,\, & {} \beta Y,\\ [Z,Y]=\,\, & {} -\beta X,\\ [W,X]=\,\, & {} b Y,\\ [W,Y]=\,\, & {} -b X,\\ [Y,X]=\,\, & {} \theta _1 Z+\theta _2 W. \end{aligned}$$

\((\mathcal {K}):\) The family is Kähler if and only if \(\theta _1=\theta _2=0\) and \(z_3=z_4=0.\) Hence we obtain the 2-dimensional family \(\mathfrak {g}_{18}^\mathcal {K}(\beta ,b)\) with

$$\begin{aligned}{}[Z,X]=\,\, & {} \beta Y,\\ [Z,Y]=\,\, & {} -\beta X,\\ [W,X]=\,\, & {} b Y,\\ [W,Y]=\,\, & {} -b X. \end{aligned}$$

The corresponding simply connected Lie group is clearly a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

Example 9.9

(\(\mathfrak {g}_{19}(\alpha ,\beta ,w_1,w_2)\)) This is a 4-dimensional family of solvable Lie algebras. Here \(\Lambda =(\alpha ,0,\beta ,0)\), which implies that

$$\begin{aligned} z_1=z_2=z_3=z_4=\theta _1=\theta _2=0 \end{aligned}$$

and the bracket relations are

$$\begin{aligned}{}[Z,X]=\,\, & {} \alpha X +\beta Y+w_1 W,\\ [Z,Y]=\,\, & {} -\beta X+\alpha Y+w_2 W. \end{aligned}$$

\((\mathcal {AK}):\) Here the almost Kähler condition is never satisfied since \(\alpha \ne 0\) and \(\theta _2=0\), hence \(\mathfrak {g}_{19}^{\mathcal {AK}}=\{0\}\).

\((\mathcal {I}):\) Because \(z_1=z_2=z_3=z_4=0\), the structure J is integrable if and only if \(w_1=w_2=0.\) We yield the 2-dimensional family \(\mathfrak {g}_{19}^\mathcal {H}(\alpha ,\beta )\) given by

$$\begin{aligned}{}[Z,X]=\,\, & {} \alpha X +\beta Y,\\ [Z,Y]=\,\, & {} -\beta X+\alpha Y. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

\((\mathcal {K}):\) The Kähler condition is never satisfied, so \(\mathfrak {g}_{19}^{\mathcal {K}}=\emptyset\).

Example 9.10

(\(\mathfrak {g}_{20}(\alpha ,a,\beta ,w_1,w_2)\)) Here we have a 5-dimensional family of solvable Lie algebras. Now \(\Lambda =(\alpha ,a,\beta ,b)\), which gives

$$\begin{aligned} z_1= & {} -\frac{aw_1}{\alpha },\ \ z_2=-\frac{aw_2}{\alpha },\ \ z_3=-\frac{a^2w_1}{\alpha ^2},\ \ z_4=-\frac{a^2w_2}{\alpha ^2}, \\ b= & {} \frac{\beta a}{\alpha },\ \ \theta _1=0,\ \ \theta _2=0. \end{aligned}$$

The bracket relations fulfill

$$\begin{aligned}{}[Z,X]=\,\, & {} \alpha X+\beta Y-\frac{aw_1}{\alpha }Z+w_1 W,\\ [Z,Y]=\,\, & {} -\beta X+\alpha Y-\frac{aw_2}{\alpha }Z+w_2 W,\\ [W,X]=\,\, & {} a X+\frac{\beta a}{\alpha }Y-\frac{a^2w_1}{\alpha ^2} Z+\frac{a}{\alpha }w_1 W,\\ [W,Y]= & {} -\frac{\beta a}{\alpha }X+a Y-\frac{a^2w_2}{\alpha ^2} Z+\frac{a}{\alpha }w_2 W. \end{aligned}$$

\((\mathcal {AK}):\) Since \(\theta _1=\theta _2=0\), \(\alpha \ne 0\) and \(a\ne 0\), the almost Kähler condition is never satisfied, hence \(\mathfrak {g}_{20}^{\mathcal {AK}}=\emptyset\).

\((\mathcal {I}):\) Here the coefficients \(z_1,\dots ,z_4,\) \(w_1\) and \(w_2\) are the same as for the family \(\mathfrak {g}_{17}\). Thus the structure J is integrable if and only if \(w_1=w_2=0\). This gives the 3-dimensional family \(\mathfrak {g}_{20}^\mathcal {I}(\alpha ,a,\beta )\) with

$$\begin{aligned}{}[Z,X]=\,\, & {} \alpha X+\beta Y,\\ [Z,Y]=\,\, & {} -\beta X+\alpha Y,\\ [W,X]=\,\, & {} a X+\frac{\beta a}{\alpha }Y,\\ [W,Y]=\,\, & {} -\frac{\beta a}{\alpha }X+a Y. \end{aligned}$$

Here the corresponding simply connected Lie group is a semidirect product \({\mathbb {R}}^2\ltimes {\mathbb {R}}^2\).

\((\mathcal {K}):\) The Kähler condition is never satisfied, therefore \(\mathfrak {g}_{20}^{\mathcal {K}}=\emptyset\).