On certain classes of Sp(4,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{Sp}(4,\mathbb{R})$$\end{document} symmetric G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2$$\end{document} structures

We find two different families of Sp(4,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{Sp}(4,\mathbb{R})$$\end{document} symmetric G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2$$\end{document} structures in seven dimensions. These are G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2$$\end{document} structures with G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2$$\end{document} being the split real form of the simple exceptional complex Lie group G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2$$\end{document}. The first family has τ2≡0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _2\equiv 0$$\end{document}, while the second family has τ1≡τ2≡0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _1\equiv \tau _2\equiv 0$$\end{document}, where τ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _1$$\end{document}, τ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _2$$\end{document} are the celebrated G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2$$\end{document}-invariant parts of the intrinsic torsion of the G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2$$\end{document} structure. The families are different in the sense that the first one lives on a homogeneous space Sp(4,R)/SL(2,R)l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_l$$\end{document}, and the second one lives on a homogeneous space Sp(4,R)/SL(2,R)s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_s$$\end{document}. Here SL(2,R)l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{SL}(2,\mathbb{R})_l$$\end{document} is an SL(2,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{SL}(2,\mathbb{R})$$\end{document} corresponding to the sl(2,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{sl}(2,\mathbb{R})$$\end{document} related to the long roots in the root diagram of sp(4,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{sp}(4,\mathbb{R})$$\end{document}, and SL(2,R)s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{SL}(2,\mathbb{R})_s$$\end{document} is an SL(2,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{SL}(2,\mathbb{R})$$\end{document} corresponding to the sl(2,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{sl}(2,\mathbb{R})$$\end{document} related to the short roots in the root diagram of sp(4,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{sp}(4,\mathbb{R})$$\end{document}.

My immediate answer was: 'I can think about it, but I have to know which of the (2, ℝ) subgroups of (4, ℝ) I shall use to built M.' The reason for the 'but' word in my answer was that there are at least two (2, ℝ) subgroups of (4, ℝ) , which lie quite differently in there. One can see them in the root diagram above: the first (2, ℝ) corresponds to the long roots, as, for example, E 1 and E 10 , whereas the second one corresponds to the short roots, as, for example, E 2 and E 9 . Since Maciej never told me which (2, ℝ) he wants, I decided to consider both of them and to determine what kind of G 2 structures one can associate with the respective choice of a subgroup.
I emphasize that in the below considerations I will use the split real form of the simple exceptional Lie group G 2 . Therefore, the corresponding G 2 structure metrics will not be Riemannian. 1 They will have signature (3,4). (4, ℝ) The Lie algebra (4, ℝ) is given by the 4 × 4 matrices where the coefficients a I , I = 1, 2, … 10 , are real constants. The Lie bracket in (4, ℝ) is the usual commutator [E,

The Lie algebra
⋅ E of two matrices E and E ′ . We start with the following basis (E I ), in (4, ℝ).
In this basis, modulo the antisymmetry, we have the following nonvanishing commutators: [E 1 , a 5 a 7 a 9 2a 10 −a 4 a 6 a 8 a 9 a 2 a 3 −a 6 −a 7 −2a 1 a 2 a 4 −a 5 We see that there are at least two (2, ℝ) Lie algebras here. The first one is and the second is The reason for distinguishing these two is as follows: The eight 1-dimensional vector subspaces I = Span(E I ) , I = 1, 2, 3, 4, 7, 8, 9, 10 , of (4, ℝ) are the root spaces of this Lie algebra. They correspond to the Cartan subalgebra of (4, ℝ) given by = Span(E 5 , E 6 ) . It follows that the pairs (E I , E J ) of the root vectors, such that I + J = 11 , I, J ≠ 5, 6 , correspond to the opposite roots of (2, ℝ) . Knowing the Killing form for (2, ℝ) , which in the basis (E I ) , and its dual basis (E I ) , , is one can see that the roots corresponding to the root vectors (E 1 , E 10 ) and (E 3 , E 8 ) are long, and the roots corresponding to the root vectors (E 2 , E 9 ) and (E 4 , E 7 ) are short; compare the Euclidian lengths of these roots as drawn on the G 2 root diagram presented at the beginning of this article. 2 Thus, the Lie algebra (2, ℝ) l containing root vectors (E 1 , E 10 ) corresponding to the long roots lies quite different in (4, ℝ) than the Lie algebra (2, ℝ) s containing the root vectors (E 2 , E 9 ) corresponding to the short roots.

Compatible pairs (g, ) on M l
To consider the homogeneous space M l = (4, ℝ)∕ (2, ℝ) l , it is convenient to change the basis (E I ) in (4, ℝ) to a new one, (e I ) , in which the last three vectors span (2, ℝ) l . Thus, we take: If now, one considers (e I ) as the basis of the Lie algebra of left invariant vector fields on the Lie group (4, ℝ) then the dual basis (e I ) , , of the left invariant forms on (4, ℝ) satisfies: 6 , e 5 = E 7 , e 6 = E 8 , e 7 = E 9 , e 8 = E 1 , e 9 = E 5 , e 10 = E 10 .
In this section, I will use from now on Greek indices , , etc., to run from 1 to 7. They number the first seven basis elements in the bases (e I ) and (e I ).
Now, I look for all bilinear symmetric forms g = g e ⊙ e on (4, ℝ) , with constant coefficients g = g , which are constant along the leaves of the foliation defined by D l . Technically, I search for those g whose Lie derivative with respect to any vector field X from D l vanishes, I have the following proposition: Thus, I have a 7-parameter family of bilinear forms on (4, ℝ) that descend to welldefined pseudo-Riemannian metrics on the leaf space M l . Note that the restriction of the Killing form K to the space where (e 8 , e 9 , e 10 ) ≡ 0 is in this family. This corresponds to g 22 = g 24 = g 46 = 0 and g 44 = 2g 26 = −g 35 = 1.
Since the aim of my note is not to be exhaustive, but rather to show how to produce G 2 structures on (4, ℝ) homogeneous spaces, from now on I will restrict myself to only one (2, ℝ) l invariant bilinear form g on (4, ℝ) , namely to coming from the restriction of the Killing form. It follows from Proposition 3.1 that this form is a well-defined (3,4) signature metric on the quotient space M l = (4, ℝ)∕ (2, ℝ) l .
I now look for the 3-forms = 1 6 e ∧ e ∧ e on (4, ℝ) that are constant along the leaves of the distribution D l , i.e., such that Then, I have the following proposition. Here e = e ∧ e ∧ e , and a, b, f, h, p, q, r, s, t and u are real constants.
Thus there is a 10-parameter family of 3-forms that descends from (4, ℝ) to the (4, ℝ) homogeneous space M l = (4, ℝ)∕ (2, ℝ) l . Now, I introduce an important notion of compatibility of a pair (g, ) where g is a metric and is a 3-form on a 7-dimensional oriented manifold M. The pair (g, ) on M is compatible if and only if Here vol(g) is a volume form on M related to the metric g. Restricting, as I did, to the (4, ℝ) invariant metric g K on M l as in (3.3), I now ask which of the 3-forms from Proposition 3.2 are compatible with the metric (3.3). In other words, I now look for the constants a, b, f, h, p, q, r, s, t and u such that for g = g K given in (3.3).
I have the following proposition.

Proposition 3.3
The general solution to the Eq. (3.5) is given by   Here a ≠ 0 , p ≠ 0 , q ≠ 1 are free parameters, and e = e ∧ e ∧ e as before.

G 2 structures in general
Compatible pairs (g, ) on 7-dimensional manifolds are interesting since they give examples of G 2 structures [2]. In general, a G 2 structure consists of a compatible pair (g, ) of a metric g and a 3-form on a 7-dimensional manifold M, and it is in addition assumed that the 3-form is generic, meaning that at every point of M it lies in one of the two open orbits of the natural action of (7, ℝ) on 3-forms in ℝ 7 . The simple exceptional Lie group G 2 appears here as the common stabilizer in (7, ℝ) of both g and . It follows (from compatibility) that the G 2 structures can have metrics g of only two signatures: the Riemannian ones and (3, 4) signature ones. If the signature of g is Riemannian, the corresponding G 2 structure is related to the compact real form of the simple exceptional complex Lie group G 2 , and in the (3, 4) signature case the corresponding G 2 structure is related to the noncompact (split) real form of the complex group G 2 . In this sense, our Corollary 3.4 provides a 3-parameter family of split real form G 2 structures on M l . G 2 structures can be classified according to the properties of their intrinsic torsion [1,2]. Making a long story short, this torsion is totally determined by finding four p-forms p on M, p = 0, 1, 2, 3 , each belonging to one of four different irreducible representations of G 2 . Before telling on how to find these forms given a G 2 structure (g, ) , we need some preparation.
We recall that the group G 2 acts in ℝ 7 , and this induces its action on spaces ⋀ p of p-forms in ℝ 7 . Of course the 1-dimensional space ⋀ 0 is G 2 irreducible, as well as is the space of 1-forms ⋀ 1 = ⋀ 1 7 . The G 2 irreducible decompositions of the spaces of 2-and 3-forms look like 27 . Here we use the convention that the lower index i in ⋀ p i denotes the dimension of the corresponding representation. It is further convenient to introduce the Hodge dual, * , which is defined on p-forms by 7 are all G 2 equivalent. Also, one can see that, e.g., The intrinsic torsion components 0 , 1 , 2 and 3 have values in the following G 2 irreducible modules: the 3-form 3 has values in the 27-dimensional irreducible representation ⋀ 3 27 , the 2-form 2 has values in the 14-dimensional irreducible representation ⋀ 2 14 , the 1-form 1 has values in the 7-dimensional irreducible representation ⋀ 1 7 , and the 0-form 0 has values in the trivial representation ⋀ 0 1 . The result of Bryant [1,2] states that for every G 2 structure (g, ) on M there exist unique forms 0 , 1 , 2 and 3 on M, with values in the above-mentioned representations, such that Thus, Eq. (3.6) enable to determine all the intrinsic torsion components 0 , 1 , 2 and 3 of a given G 2 structure (g, ) . They are called Bryant's [1,2] equations. It follows that vanishing or not of each of the forms p is a G 2 invariant property of a G 2 structure.

All (4, ℝ) symmetric G 2 structures on M l with the metric coming from the Killing form
The below theorem characterizes the G 2 structures corresponding to compatible pairs (g K , ) from Corollary 3.4; it summarizes the already obtained results and, in addition, provides formulas for the intrinsic torsion which are needed for the characterization.
where as usual e = e ∧ e and e = e ∧ e ∧ e .
Thus, the 3-parameter family of G 2 structures on M l described in this theorem have the entire 14-dimensional torsion 2 = 0 . This means that all these G 2 structures are integrable in the terminology of [3,4], or what is the same, this means that they all have the totally skew symmetric torsion.

G 2 structures on (4, ℝ)∕ (2, ℝ) s
Now we consider the homogeneous space M s = (4, ℝ)∕ (2, ℝ) s . Since (2, ℝ) is spanned by E 2 , E 5 + E 6 , E 9 , it is convenient to put these vectors at the end of the new basis of the Lie algebra (4, ℝ) . We choose this new basis (f I ) in (4, ℝ) as: If now, one considers (f I ) as the basis of the Lie algebra of invariant vector fields on the Lie group (4, ℝ) , then the dual basis (f I ) , , of the left invariant forms on (4, ℝ) , satisfies: In this basis, the Killing form on (4, ℝ) is where as usual the structure constants c I JK are defined by [f I , f J ] = c K IJ f K . Using the same arguments, as in the case of M l , we again see that (4, ℝ) has the structure of the principal (2, ℝ) fiber bundle (2, ℝ) s → (4, ℝ) → M s = (4, ℝ)∕ (2, ℝ) s over the homogeneous space M s = (4, ℝ)∕ (2, ℝ) s . In particular, we have a foliation of (4, ℝ) by integral leaves of an integrable distribution D s spanned by the annihilator of the forms (f 1 , f 2 , … , f 7 ) . As before, also in this section, we will use Greek indices , , etc., to run from 1 to 7. They now number the first seven basis elements in the bases (f I ) and (f I ).
Repeating the procedure from the previous sections, I now search for all bilinear symmetric forms g = g f ⊙ f on (4, ℝ) , with constant coefficients g = g , whose Lie derivative with respect to any vector field X from D l vanishes, I have the following proposition. (4.1)
Again for simplicity reasons, I will solve the problem of finding (4, ℝ) invariant G 2 structures on M s restricting to only those pairs (g, ) for which g = g K , where which again means that I only will consider one metric, the one coming from the restriction of the Killing form of (4, ℝ) to M s . It is a well-defined (3,4)   f ∧ f ∧ f on (4, ℝ) which satisfies condition (4.4). The general formula for is: Here f = f ∧ f ∧ f , and a, b, q, h and p are real constants.
Solving for all 3-forms from this 5-parameter family that are compatible, as in (3.5), with the metric g K from (4.3), I arrive at the following proposition.  Here q ≠ 0 is a free parameter, and f = f ∧ f ∧ f as before. constant growth vector (2,3), while the distributions on M s are integrable. These pairs of distributions constitute an immanent ingredient of the geometry on the corresponding spaces M l and M s and, since they are diffeomorphically nonequivalent and they make the G 2 geometries there quite different. I believe that this fact makes the G 2 structures obtained on M l and M s really nonequivalent.