Geodesic flows on real forms of complex semi-simple Lie groups of rigid body type

Abstract

The geodesic flows are studied on real forms of complex semi-simple Lie groups with respect to a left-invariant (pseudo-)Riemannian metric of rigid body type. The Williamson types of the isolated relative equilibria on generic adjoint orbits are determined.

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Notes

  1. 1.

    In the proof of [19, Theorem 6.94], the real form is arbitrarily chosen, but we choose a compact real form \({\mathfrak {g}}_u\).

  2. 2.

    In the first table in [29], this symbol is misprinted as \(\mathbf {EII}\) at the first case of the row \(E_6\) and the last column.

  3. 3.

    This symbol appears in the first column of Tables 1 and 2, e.g., as \(A_{\ell }\left( B_{\frac{\ell }{2}}\right) \).

  4. 4.

    There seems to be a typo at the number of focus–focus components of the Williamson type for \(\mathfrak {gl}_k\left( {\mathbb {H}}\right) \) in [18, Corollary 7.5]; the number \(2k(k-1)\) should be read as \(k(k-1)\). Then, it agrees with the data for the real simple Lie algebra of type \(\mathbf{AII }\) in Table 1 by putting \(k=\left( \ell +1\right) /2\).

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Acknowledgements

We thank Alexey Bolsinov for his interests and comments on the present work and Nobutaka Bomuki for his comment on Corollary 4.7. We also thank the referee for the valuable comments. We are grateful to Ritsumeikan University, Shanghai Jiao Tong University, and NCCR SwissMAP grant for facilitating our collaboration.

Funding

The first author is partially supported by the National Natural Science Foundation of China Grant Number 11871334 and NCCR Swiss MAP grant of the Swiss National Science Foundation. The second author is partially supported by Grant for Basic Science Research Projects from The Sumitomo Foundation and by JSPS KAKENHI Grant Number JP19K14540.

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Appendix: Computational details for the linearization

Appendix: Computational details for the linearization

Each basis element of \(T_X{\mathcal {O}}\) in (4.3) is given in the form \(\mathrm {ad}_Y^{*} X\), where Y is one of the elements (2.5). By Lemmas 2.22.32.4, the vector Y is expressed as either \(\displaystyle \frac{\eta +{\overline{\eta }}}{2}\) or \(\displaystyle \frac{\eta -{\overline{\eta }}}{2\sqrt{-1}}\), where \(\eta \in {\mathfrak {g}}_{\delta }\) for some \(\delta \in \Delta \). Recall that \({\overline{\eta }}\in {\mathfrak {g}}_{\overline{\delta }}\) by the same lemmas.

Proof of the formulas (4.4), (4.5), (4.6) Each entry of the matrix representation of \(\omega _{{\mathcal {O}}}\) in this chosen basis (4.3) of \(T_X{\mathcal {O}}\) is given by

$$\begin{aligned} \omega _{{\mathcal {O}}}\left( \mathrm {ad}_YX, \mathrm {ad}_ZX\right)&=\kappa \left( X, \left[ Y,Z\right] \right) =\kappa \left( \left[ X,Y\right] , Z\right) , \end{aligned}$$
(4.19)

where \(Y=\displaystyle \frac{\eta +{\overline{\eta }}}{2}\) or \(Y=\displaystyle \frac{\eta -{\overline{\eta }}}{2\sqrt{-1}}\) and \(Z=\displaystyle \frac{\zeta +{\overline{\zeta }}}{2}\) or \(Z=\displaystyle \frac{\zeta -{\overline{\zeta }}}{2\sqrt{-1}}\), with \(\eta \in {\mathfrak {g}}_{\delta }\), \(\zeta \in {\mathfrak {g}}_{\epsilon }\) for some \(\delta , \epsilon \in \Delta \).

For \(Y=\displaystyle \frac{\eta +{\overline{\eta }}}{2}\) and \(Z=\displaystyle \frac{\zeta +{\overline{\zeta }}}{2}\), we have

$$\begin{aligned} \omega _{{\mathcal {O}}}\left( \mathrm {ad}_YX, \mathrm {ad}_ZX\right)&=\frac{1}{4}\kappa \left( \left[ X, \eta +{\overline{\eta }}\right] , \zeta +{\overline{\zeta }}\right) =\frac{1}{4}\kappa \left( \delta (X)\eta +\overline{\delta (X)}{\overline{\eta }}, \zeta +{\overline{\zeta }}\right) \nonumber \\&=\frac{1}{4}\left( \delta (X)\left( \kappa (\eta , \zeta )+\kappa (\eta , {\overline{\zeta }})\right) +\overline{\delta (X)}\left( \kappa ({\overline{\eta }}, \zeta ) +\kappa ({\overline{\eta }}, {\overline{\zeta }})\right) \right) . \end{aligned}$$
(4.20)

Similarly, if \(Y=\displaystyle \frac{\eta +{\overline{\eta }}}{2}\) and \(Z=\displaystyle \frac{\zeta -{\overline{\zeta }}}{2\sqrt{-1}}\), we have

$$\begin{aligned} \omega _{{\mathcal {O}}}\left( \mathrm {ad}_YX, \mathrm {ad}_ZX\right) =\frac{1}{4\sqrt{-1}}\left( \delta (X)\left( \kappa (\eta , \zeta )- \kappa (\eta , {\overline{\zeta }})\right) +\overline{\delta (X)} \left( \kappa ({\overline{\eta }}, \zeta )- \kappa ({\overline{\eta }}, {\overline{\zeta }})\right) \right) .\nonumber \\ \end{aligned}$$
(4.21)

For \(Y=\displaystyle \frac{\eta -{\overline{\eta }}}{2\sqrt{-1}}\) and \(Z=\displaystyle \frac{\zeta -{\overline{\zeta }}}{2\sqrt{-1}}\), we have

$$\begin{aligned} \omega _{{\mathcal {O}}}\left( \mathrm {ad}_YX, \mathrm {ad}_ZX\right) =-\frac{1}{4}\left( \delta (X)\left( \kappa (\eta , \zeta )- \kappa (\eta , {\overline{\zeta }})\right) - \overline{\delta (X)}\left( \kappa ({\overline{\eta }}, \zeta )- \kappa ({\overline{\eta }}, {\overline{\zeta }})\right) \right) .\nonumber \\ \end{aligned}$$
(4.22)

Note that \(\kappa \left( \eta , \zeta \right) \) and \(\kappa \left( {\overline{\eta }}, {\overline{\zeta }}\right) \) are nonzero only if \(\delta +\epsilon =0\); similarly, \(\kappa \left( \eta , {\overline{\zeta }}\right) \) and \(\kappa \left( {\overline{\eta }}, \zeta \right) \) are nonzero only if \(\delta +{\overline{\epsilon }}=0\). As a result, the matrix representation of the orbit symplectic form \(\omega _{{\mathcal {O}}}\) is a block diagonal matrix whose blocks are in (4.4), (4.5), (4.6).

By the expression (3.9) of the orbit symplectic form \(\omega _{{\mathcal {O}}}\), formula (4.4) is obtained from the following computation:

$$\begin{aligned} \omega _{{\mathcal {O}}}\left( \mathrm {ad}_{u_{\alpha }}X, \mathrm {ad}_{\theta u_{\alpha }}X\right) =\kappa \left( X, \left[ u_{\alpha }, \theta u_{\alpha }\right] \right) =\kappa \left( \left[ X, u_{\alpha }\right] , \theta u_{\alpha }\right) =\alpha (X)\kappa \left( u_{\alpha }, \theta u_{\alpha }\right) . \end{aligned}$$

We used \(u_{\alpha }\in {\mathfrak {g}}_{\alpha }^{{\mathbb {C}}}\).

Similarly, formula (4.5) is obtained as follows:

$$\begin{aligned} \omega _{{\mathcal {O}}}\left( \mathrm {ad}_{v_{\beta }^r}X, \mathrm {ad}_{v_{\beta }^i}X\right)&=\kappa \left( X, \left[ v_{\beta }^r, v_{\beta }^i\right] \right) =\kappa \left( X, \left[ \frac{v_{\beta }+\overline{v_{\beta }}}{2}, \frac{v_{\beta }-\overline{v_{\beta }}}{2\sqrt{-1}}\right] \right) \\&=\frac{\sqrt{-1}}{2}\kappa \left( X, \left[ v_{\beta }, \overline{v_{\beta }}\right] \right) =\frac{\sqrt{-1}}{2}\kappa \left( \left[ X, v_{\beta }\right] , \overline{v_{\beta }}\right) \\&=\frac{\sqrt{-1}}{2}\beta (X)\kappa \left( v_{\beta }, \overline{v_{\beta }}\right) . \end{aligned}$$

Here, we used \(v_{\beta }\in {\mathfrak {g}}_{\beta }^{{\mathbb {C}}}\).

Formula (4.6) is obtained from the following computations:

$$\begin{aligned} \omega _{{\mathcal {O}}}&\left( \mathrm {ad}_{w_{\gamma }^r}X, \mathrm {ad}_{w_{\gamma }^i}X\right) \\&{\mathop {=}\limits ^{(4.21)}} \frac{1}{4\sqrt{-1}}\left( \gamma (X)\left( \kappa (w_{\gamma }, w_{\gamma })- \kappa (w_{\gamma },\overline{w_{\gamma }}\right) +\overline{\gamma (X)}\left( \kappa (\overline{w_{\gamma }}, w_{\gamma })- \kappa (\overline{w_{\gamma }},\overline{w_{\gamma }}\right) \right) =0,\\ \omega _{{\mathcal {O}}}&\left( \mathrm {ad}_{w_{\gamma }^r}X, \mathrm {ad}_{w_{-\gamma }^r}X\right) \\&{\mathop {=}\limits ^{(4.20)}}\frac{1}{4}\left( \gamma (X) \left( \kappa \left( w_{\gamma }, w_{-\gamma }\right) + \kappa \left( w_{\gamma }, \overline{w_{-\gamma }}\right) \right) \right. \\&\qquad \qquad \qquad \quad \left. + \overline{\gamma (X)}\left( \kappa \left( \overline{w_{\gamma }}, w_{-\gamma }\right) + \kappa \left( \overline{w_{\gamma }}, \overline{w_{-\gamma }}\right) \right) \right) \\&\;\;\, =\frac{1}{4}\left( \gamma (X)\kappa \left( w_{\gamma }, w_{-\gamma }\right) + \overline{\gamma (X)}\kappa \left( \overline{w_{\gamma }}, \overline{w_{-\gamma }}\right) \right) =\frac{1}{2}\mathrm {Re}\left( \gamma (X)\kappa (w_{\gamma },w_{-\gamma }) \right) , \\ \omega _{{\mathcal {O}}}&\left( \mathrm {ad}_{w_{\gamma }^r}X, \mathrm {ad}_{w_{-\gamma }^i}X\right) \\&{\mathop {=}\limits ^{(4.21)}}\frac{1}{4\sqrt{-1}} \left( \gamma (X) \left( \kappa \left( w_{\gamma }, w_{-\gamma }\right) -\kappa \left( w_{\gamma }, \overline{w_{-\gamma }}\right) \right) \right. \\&\qquad \qquad \qquad \quad \left. + \overline{\gamma (X)} \left( \kappa \left( \overline{w_{\gamma }}, w_{-\gamma }\right) - \kappa \left( \overline{w_{\gamma }}, \overline{w_{-\gamma }}\right) \right) \right) \\&\;\;\,=\frac{1}{4\sqrt{-1}} \left( \gamma (X) \kappa \left( w_{\gamma }, w_{-\gamma }\right) - \overline{\gamma (X)} \kappa \left( \overline{w_{\gamma }}, \overline{w_{-\gamma }}\right) \right) =\frac{1}{2}\mathrm {Im}\left( \gamma (X)\kappa \left( w_{\gamma }, w_{-\gamma }\right) \right) , \\ \omega _{{\mathcal {O}}}&\left( \mathrm {ad}_{w_{\gamma }^i}X, \mathrm {ad}_{w_{-\gamma }^r}X\right) \\&{\mathop {=}\limits ^{(4.21)}} \frac{1}{4\sqrt{-1}}\kappa \left( \gamma (X) \left( \kappa (w_{\gamma }, w_{-\gamma })+ \kappa (w_{\gamma }, \overline{w_{-\gamma }})\right) \right. \\&\qquad \qquad \qquad \quad \left. - \overline{\gamma (X)} \left( \kappa (\overline{w_{\gamma }}, w_{-\gamma })+\kappa (\overline{w_{\gamma }}, \overline{w_{-\gamma }})\right) \right) \\&\;\;\,=\frac{1}{4\sqrt{-1}}\kappa \left( \gamma (X) \kappa (w_{\gamma }, w_{-\gamma }) - \overline{\gamma (X)} \kappa (\overline{w_{\gamma }}, \overline{w_{-\gamma }}) \right) =\frac{1}{2}\mathrm {Im} \left( \gamma (X) \kappa (w_{\gamma }, w_{-\gamma }) \right) , \\ \omega _{{\mathcal {O}}}&\left( \mathrm {ad}_{w_{\gamma }^i}X, \mathrm {ad}_{w_{-\gamma }^i}X\right) \\&{\mathop {=}\limits ^{(4.22)}} -\frac{1}{4}\left( \gamma (X)\left( \kappa (w_{\gamma }, w_{-\gamma })-\kappa (w_{\gamma }, \overline{w_{-\gamma }})\right) \right. \\&\qquad \qquad \qquad \quad \left. - \overline{\gamma (X)}\left( \kappa (\overline{w_{\gamma }}, w_{-\gamma })- \kappa (\overline{w_{\gamma }}, \overline{w_{-\gamma }})\right) \right) \\&\;\;\,=-\frac{1}{4}\left( \gamma (X)\kappa (w_{\gamma }, w_{-\gamma }) +\overline{\gamma (X)}\kappa (\overline{w_{\gamma }}, \overline{w_{-\gamma }}) \right) =-\frac{1}{2}\mathrm {Re}\left( \gamma (X)\kappa (w_{\gamma }, w_{-\gamma })\right) , \\&\omega _{{\mathcal {O}}}\left( \mathrm {ad}_{w_{-\gamma }^r}X, \mathrm {ad}_{w_{-\gamma }^i}X\right) \\&{\mathop {=}\limits ^{(4.21)}} \frac{1}{4\sqrt{-1}}\left( -\gamma (X)\left( \kappa (w_{-\gamma }, w_{-\gamma })- \kappa (w_{-\gamma }, \overline{w_{-\gamma }})\right) \right. \\&\qquad \qquad \qquad \quad \left. - \overline{\gamma (X)}\left( \kappa (\overline{w_{-\gamma }}, w_{-\gamma })- \kappa (\overline{w_{-\gamma }}, \overline{w_{-\gamma }})\right) \right) =0. \end{aligned}$$

Here, we used the consequences of Lemma 2.4: \(w_{\gamma }\in {\mathfrak {g}}_{\gamma }^{{\mathbb {C}}}\), \(\overline{w_{\gamma }}\in {\mathfrak {g}}_{{\overline{\gamma }}}^{{\mathbb {C}}}\), \({\mathfrak {g}}_{\gamma }\perp _{\kappa }{\mathfrak {g}}_{\gamma }\), \({\mathfrak {g}}_{\pm \gamma }\perp _{\kappa } {\mathfrak {g}}_{\pm {\overline{\gamma }}}\), where all different signs can be taken.

Proof of formulas (4.11), (4.12), (4.13) Each entry of the matrix representation of \(\mathrm {Hess}_X\left[ H|_{{\mathcal {O}}}\right] \) in the basis (4.3) of \(T_X{\mathcal {O}}\) is given by (4.10), where \(Y=\displaystyle \frac{\eta +{\overline{\eta }}}{2}\) or \(Y=\displaystyle \frac{\eta -{\overline{\eta }}}{2\sqrt{-1}}\) and \(Z=\displaystyle \frac{\zeta +{\overline{\zeta }}}{2}\) or \(Z=\displaystyle \frac{\zeta -{\overline{\zeta }}}{2\sqrt{-1}}\), with \(\eta \in {\mathfrak {g}}_{\delta }\), \(\zeta \in {\mathfrak {g}}_{\epsilon }\) for some \(\delta , \epsilon \in \Delta \).

For \(Y=\displaystyle \frac{\eta +{\overline{\eta }}}{2}\) and \(Z=\displaystyle \frac{\zeta +{\overline{\zeta }}}{2}\), we have

$$\begin{aligned} \kappa&\left( \mathrm {ad}_YX, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}}\right] \left( \mathrm {ad}_ZX\right) \right) =\kappa \left( \mathrm {ad}_{(\eta +{\overline{\eta }})/2}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}}\right] \left( \mathrm {ad}_{(\zeta +{\overline{\zeta }})/2}X\right) \right) \nonumber \\&=\frac{1}{4}\kappa \left( \mathrm {ad}_{(\eta +{\overline{\eta }})}X, \varphi _{a,b}\left( \mathrm {ad}_{(\zeta +{\overline{\zeta }})}X\right) - \mathrm {ad}_{(\zeta +{\overline{\zeta }})}\left( D(X)\right) \right) \nonumber \\&=\frac{1}{4}\kappa \left( \delta (X)\eta +\overline{\delta (X)}{\overline{\eta }}, \epsilon (X)\varphi _{a,b}(\zeta )+\overline{\epsilon (X)}\varphi _{a,b} ({\overline{\zeta }})-\epsilon (D(X))\zeta -\overline{\epsilon (D(X))} {\overline{\zeta }}\right) \nonumber \\&=\frac{1}{4}\kappa \left( \delta (X)\eta +\overline{\delta (X)}{\overline{\eta }}, \epsilon (X)\frac{\epsilon (b)}{\epsilon (a)}\zeta +\overline{\epsilon (X)} \frac{\overline{\epsilon (b)}}{\overline{\epsilon (a)}}{\overline{\zeta }}- \epsilon (D(X))\zeta -\overline{\epsilon (D(X))}{\overline{\zeta }}\right) \nonumber \\&=\frac{1}{4}\kappa \left( \delta (X)\eta +\overline{\delta (X)}{\overline{\eta }}, \left( \epsilon (X)\frac{\epsilon (b)}{\epsilon (a)}-\epsilon (D(X))\right) \zeta +\overline{\left( \epsilon (X)\frac{\epsilon (b)}{\epsilon (a)}- \epsilon (D(X))\right) }{\overline{\zeta }}\right) \nonumber \\&=\frac{1}{4} \left( \delta (X)\left( \epsilon (X)\frac{\epsilon (b)}{\epsilon (a)}- \epsilon (D(X))\right) \kappa (\eta ,\zeta ) +\delta (X)\overline{\left( \epsilon (X)\frac{\epsilon (b)}{\epsilon (a)}- \epsilon (D(X))\right) }\kappa (\eta ,{\overline{\zeta }})\right. \nonumber \\&\qquad \left. +\overline{\delta (X)}\left( \epsilon (X)\frac{\epsilon (b)}{\epsilon (a)}-\epsilon (D(X))\right) \kappa ({\overline{\eta }},\zeta ) +\overline{\delta (X)}\overline{\left( \epsilon (X) \frac{\epsilon (b)}{\epsilon (a)}- \epsilon (D(X))\right) }\kappa ({\overline{\eta }},{\overline{\zeta }}) \right) . \end{aligned}$$
(4.23)

Similarly, if \(Y=\displaystyle \frac{\eta +{\overline{\eta }}}{2}\) and \(Z=\displaystyle \frac{\zeta -{\overline{\zeta }}}{2\sqrt{-1}}\), we have

$$\begin{aligned} \kappa&\left( \mathrm {ad}_YX, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}}\right] \left( \mathrm {ad}_ZX\right) \right) =\kappa \left( \mathrm {ad}_{(\eta +{\overline{\eta }})/2}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}}\right] \left( \mathrm {ad}_{(\zeta -{\overline{\zeta }})/2\sqrt{-1}}X\right) \right) \nonumber \\&=\frac{1}{4\sqrt{-1}} \left( \delta (X)\left( \epsilon (X)\frac{\epsilon (b)}{\epsilon (a)}- \epsilon (D(X))\right) \kappa (\eta ,\zeta )\right. \nonumber \\&\qquad \qquad \qquad \left. -\delta (X)\overline{\left( \epsilon (X)\frac{\epsilon (b)}{\epsilon (a)}- \epsilon (D(X))\right) }\kappa (\eta ,{\overline{\zeta }})\right. \nonumber \\&\qquad \left. +\overline{\delta (X)}\left( \epsilon (X)\frac{\epsilon (b)}{\epsilon (a)}-\epsilon (D(X))\right) \kappa ({\overline{\eta }},\zeta ) -\overline{\delta (X)}\overline{\left( \epsilon (X) \frac{\epsilon (b)}{\epsilon (a)}- \epsilon (D(X))\right) }\kappa ({\overline{\eta }},{\overline{\zeta }}) \right) . \end{aligned}$$
(4.24)

If \(Y=\displaystyle \frac{\eta -{\overline{\eta }}}{2\sqrt{-1}}\) and \(Z=\displaystyle \frac{\zeta -{\overline{\zeta }}}{2\sqrt{-1}}\), we have

$$\begin{aligned} \kappa&\left( \mathrm {ad}_YX, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}}\right] \left( \mathrm {ad}_ZX\right) \right) =\kappa \left( \mathrm {ad}_{(\eta -{\overline{\eta }})/2\sqrt{-1}}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}}\right] \left( \mathrm {ad}_{(\zeta -{\overline{\zeta }})/2\sqrt{-1}}X\right) \right) \nonumber \\&=-\frac{1}{4}\left( \delta (X)\left( \epsilon (X)\frac{\epsilon (b)}{\epsilon (a)}- \epsilon (D(X))\right) \kappa (\eta ,\zeta ) -\delta (X)\overline{\left( \epsilon (X)\frac{\epsilon (b)}{\epsilon (a)}- \epsilon (D(X))\right) }\kappa (\eta ,{\overline{\zeta }})\right. \nonumber \\&\qquad \left. -\overline{\delta (X)}\left( \epsilon (X)\frac{\epsilon (b)}{\epsilon (a)}-\epsilon (D(X))\right) \kappa ({\overline{\eta }},\zeta ) +\overline{\delta (X)}\overline{\left( \epsilon (X) \frac{\epsilon (b)}{\epsilon (a)}- \epsilon (D(X))\right) }\kappa ({\overline{\eta }},{\overline{\zeta }}) \right) . \end{aligned}$$
(4.25)

In the above computations, we used the formulas

$$\begin{aligned} \varphi _{a,b}\left( \eta \right) = \displaystyle \frac{\alpha (b)}{\alpha (a)}\eta , \qquad \varphi _{a,b}\left( {\overline{\eta }}\right) = \displaystyle \frac{\overline{\alpha (b)}}{\overline{\alpha (a)}} {\overline{\eta }}, \qquad \varphi _{a,b}\left( \zeta \right) = \displaystyle \frac{\alpha (b)}{\alpha (a)}\zeta , \qquad \varphi _{a,b}\left( {\overline{\zeta }}\right) = \displaystyle \frac{\overline{\alpha (b)}}{\overline{\alpha (a)}} {\overline{\zeta }}, \end{aligned}$$

which follow from \(\eta \in {\mathfrak {g}}_{\delta }\), \({\overline{\eta }}\in {\mathfrak {g}}_{{\overline{\delta }}}\), \(\zeta \in {\mathfrak {g}}_{\epsilon }\), \({\overline{\zeta }}\in {\mathfrak {g}}_{{\overline{\epsilon }}}\).

Note that \(\kappa \left( \eta , \zeta \right) \) and \(\kappa \left( {\overline{\eta }}, {\overline{\zeta }}\right) \) are nonzero only if \(\delta +\epsilon =0\) and \(\kappa \left( \eta , {\overline{\zeta }}\right) \) and \(\kappa \left( {\overline{\eta }}, \zeta \right) \) are nonzero only if \(\delta +{\overline{\epsilon }}=0\). As a result, the matrix representation of the Hessian \(\mathrm {Hess}_X\left[ H|_{{\mathcal {O}}}\right] \) is a block diagonal matrix whose blocks are given in formulas (4.11), (4.12), (4.13).

By the expression (4.10) of the Hessian \(\mathrm {Hess}_X\left[ H|_{{\mathcal {O}}}\right] \), formula (4.11) is obtained from the following computations:

$$\begin{aligned} \kappa \left( \mathrm {ad}_{u_{\alpha }}X, \varphi _{a,b} \left( \mathrm {ad}_{u_{\alpha }}X\right) -\mathrm {ad}_{u_{\alpha }}D(X)\right)&=\kappa \left( -\alpha (X) u_{\alpha }, -\varphi _{a,b}\left( \alpha (X) u_{\alpha }\right) +\alpha (D(X)) u_{\alpha }\right) \\&=\kappa \left( -\alpha (X) u_{\alpha }, -\alpha (X) \frac{\alpha (b)}{\alpha (a)} u_{\alpha }+\alpha (D(X)) u_{\alpha }\right) \\&=\alpha (X)\left( \alpha (X)\frac{\alpha (b)}{\alpha (a)}-\alpha (D(X))\right) \kappa \left( u_{\alpha }, u_{\alpha }\right) \\&=0, \\ \kappa \left( \mathrm {ad}_{u_{\alpha }}X, \varphi _{a,b} \left( \mathrm {ad}_{\theta u_{\alpha }}X\right) - \mathrm {ad}_{\theta u_{\alpha }}D(X)\right)&=\kappa \left( -\alpha (X) u_{\alpha }, \varphi _{a,b}\left( \alpha (X) \theta u_{\alpha }\right) -\alpha (D(X)) \theta u_{\alpha }\right) \\&=\kappa \left( -\alpha (X) u_{\alpha }, \alpha (X) \frac{\alpha (b)}{\alpha (a)} \theta u_{\alpha }-\alpha (D(X)) \theta u_{\alpha }\right) \\&=-\alpha (X)\left( \alpha (X)\frac{\alpha (b)}{\alpha (a)}-\alpha (D(X))\right) \kappa \left( u_{\alpha }, \theta u_{\alpha }\right) , \\ \kappa \left( \mathrm {ad}_{u_{\alpha }}\theta X, \varphi _{a,b}\left( \mathrm {ad}_{\theta u_{\alpha }}X\right) - \mathrm {ad}_{\theta u_{\alpha }}D(X)\right)&=\kappa \left( \alpha (X) \theta u_{\alpha }, \varphi _{a,b}\left( \alpha (X) \theta u_{\alpha }\right) -\alpha (D(X)) \theta u_{\alpha }\right) \\&=\kappa \left( \alpha (X) \theta u_{\alpha }, \alpha (X) \frac{\alpha (b)}{\alpha (a)} \theta u_{\alpha }-\alpha (D(X)) \theta u_{\alpha }\right) \\&=\alpha (X)\left( \alpha (X)\frac{\alpha (b)}{\alpha (a)}-\alpha (D(X))\right) \kappa \left( \theta u_{\alpha }, \theta u_{\alpha }\right) \\&=0. \end{aligned}$$

Here, we used the facts \(u_{\alpha }\in {\mathfrak {g}}_{\alpha }^{{\mathbb {C}}}\), \(\theta u_{\alpha }\in {\mathfrak {g}}_{-\alpha }^{{\mathbb {C}}}\), \({\mathfrak {g}}_{\alpha }\perp _{\kappa }{\mathfrak {g}}_{\alpha }\), \({\mathfrak {g}}_{-\alpha }\perp _{\kappa }{\mathfrak {g}}_{-\alpha }\) and the formulas

$$\begin{aligned} \varphi _{a,b}\left( u_{\alpha }\right) = \displaystyle \frac{\alpha (b)}{\alpha (a)}u_{\alpha }, \qquad \varphi _{a,b}\left( \theta u_{\alpha }\right) = \displaystyle \frac{\alpha (b)}{\alpha (a)}\theta u_{\alpha }, \end{aligned}$$

which follow from (3.1) and (3.2).

Similarly, formula (4.12) is obtained by

$$\begin{aligned} \kappa&\left( \mathrm {ad}_{v_{\beta }^r}X, \varphi _{a,b}\left( \mathrm {ad}_{v_{\beta }^r}X\right) -\mathrm {ad}_{v_{\beta }^r}D(X)\right) \\&=\kappa \left( -\beta (X) \sqrt{-1}v_{\beta }^i, -\varphi _{a,b}\left( \beta (X) \sqrt{-1}v_{\beta }^i\right) +\beta (X) \sqrt{-1}v_{\beta }^i\right) \\&=\kappa \left( -\beta (X) \sqrt{-1}v_{\beta }^i, -\beta (X) \sqrt{-1} \frac{\beta (b)}{\beta (a)}v_{\beta }^i+\beta (X) \sqrt{-1}v_{\beta }^i\right) \\&=-\beta (X)\left( \beta (X)\frac{\beta (b)}{\beta (a)}-\beta (D(X))\right) \kappa \left( v_{\beta }^i, v_{\beta }^i\right) \\&=\frac{1}{4}\beta (X)\left( \beta (X)\frac{\beta (b)}{\beta (a)} -\beta (D(X))\right) \kappa \left( v_{\beta }-\overline{v_{\beta }}, v_{\beta }-\overline{v_{\beta }}\right) \\&=\frac{1}{4}\beta (X)\left( \beta (X)\frac{\beta (b)}{\beta (a)} -\beta (D(X))\right) \left( \kappa \left( v_{\beta },v_{\beta }\right) -2\kappa \left( v_{\beta }, \overline{v_{\beta }}\right) +\kappa \left( \overline{v_{\beta }}, \overline{v_{\beta }}\right) \right) \\&=-\frac{1}{2}\beta (X)\left( \beta (X)\frac{\beta (b)}{\beta (a)} -\beta (D(X))\right) \kappa \left( v_{\beta }, \overline{v_{\beta }}\right) , \\ \kappa&\left( \mathrm {ad}_{v_{\beta }^r}X, \varphi _{a,b}\left( \mathrm {ad}_{v_{\beta }^i}X\right) -\mathrm {ad}_{v_{\beta }^i}D(X)\right) \\&=\kappa \left( -\beta (X)\sqrt{-1}v_{\beta }^i, \varphi _{a,b}\left( \beta (X)\sqrt{-1} v_{\beta }^r\right) -\beta (D(X))\sqrt{-1}v_{\beta }^r\right) \\&=\kappa \left( -\beta (X)\sqrt{-1}v_{\beta }^i, \beta (X)\sqrt{-1} \frac{\beta (b)}{\beta (a)}v_{\beta }^r-\beta (D(X))\sqrt{-1}v_{\beta }^r\right) \\&=\beta (X)\left( \beta (X)\frac{\beta (b)}{\beta (a)}-\beta (D(X))\right) \kappa \left( v_{\beta }^i, v_{\beta }^r\right) \\&=\frac{1}{4\sqrt{-1}}\beta (X)\left( \beta (X)\frac{\beta (b)}{\beta (a)}- \beta (D(X))\right) \kappa \left( v_{\beta }-\overline{v_{\beta }}, v_{\beta }+\overline{v_{\beta }}\right) \\&=\frac{1}{4\sqrt{-1}}\beta (X)\left( \beta (X)\frac{\beta (b)}{\beta (a)}- \beta (D(X))\right) \left( \kappa \left( v_{\beta },v_{\beta }\right) + \kappa \left( \overline{v_{\beta }}, \overline{v_{\beta }}\right) \right) =0, \\ \kappa&\left( \mathrm {ad}_{v_{\beta }^i}X, \varphi _{a,b}\left( \mathrm {ad}_{v_{\beta }^i}X\right) -\mathrm {ad}_{v_{\beta }^i}D(X)\right) \\&=\kappa \left( \beta (X)\sqrt{-1}v_{\beta }^r, \varphi _{a,b}\left( \beta (X) \sqrt{-1}v_{\beta }^r\right) -\beta (D(X))\sqrt{-1}v_{\beta }^r\right) \\&=\kappa \left( \beta (X)\sqrt{-1}v_{\beta }^i, \beta (X)\sqrt{-1} \frac{\beta (b)}{\beta (a)}v_{\beta }^r-\beta (D(X))\sqrt{-1}v_{\beta }^r\right) \\&=-\beta (X)\left( \beta (X)\frac{\beta (b)}{\beta (a)}-\beta (D(X))\right) \kappa \left( v_{\beta }^r, v_{\beta }^r\right) \\&=-\frac{1}{4}\beta (X)\left( \beta (X)\frac{\beta (b)}{\beta (a)} -\beta (D(X))\right) \kappa \left( v_{\beta }+\overline{v_{\beta }}, v_{\beta }+\overline{v_{\beta }}\right) \\&=-\frac{1}{4}\beta (X)\left( \beta (X)\frac{\beta (b)}{\beta (a)} -\beta (D(X))\right) \left( \kappa \left( v_{\beta },v_{\beta }\right) +2\kappa \left( v_{\beta }, \overline{v_{\beta }}\right) +\kappa \left( \overline{v_{\beta }}, \overline{v_{\beta }}\right) \right) \\&=-\frac{1}{2}\beta (X)\left( \beta (X)\frac{\beta (b)}{\beta (a)} -\beta (D(X))\right) \kappa \left( v_{\beta }, \overline{v_{\beta }}\right) . \end{aligned}$$

Here, we used the fact \(v_{\beta }\in {\mathfrak {g}}_{\beta }^{{\mathbb {C}}}\), \(\overline{v_{\beta }}\in {\mathfrak {g}}_{-\beta }^{{\mathbb {C}}}\), \({\mathfrak {g}}_{\beta }\perp _{\kappa }{\mathfrak {g}}_{\beta }\), \({\mathfrak {g}}_{-\beta }\perp _{\kappa }{\mathfrak {g}}_{-\beta }\) and the formulas

$$\begin{aligned} \varphi _{a,b}\left( v_{\beta }\right) = \displaystyle \frac{\beta (b)}{\beta (a)}v_{\beta }, \qquad \varphi _{a,b}\left( \overline{v_{\beta }}\right) = \displaystyle \frac{\beta (b)}{\beta (a)}\overline{v_{\beta }}, \end{aligned}$$

which follow from (3.1) and (3.2).

Formula (4.13) is obtained through the following computations:

$$\begin{aligned} \kappa&\left( \mathrm {ad}_{w_{\gamma }^r}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}}\right] \left( \mathrm {ad}_{w_{\gamma }^r}X\right) \right) \\&{\mathop {=}\limits ^{(4.23)}} \frac{1}{4} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{\gamma })\right. \\&\qquad \left. +\gamma (X)\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X)) \right) }\kappa (w_{\gamma },\overline{w_{\gamma }})\right. \\&\qquad \left. +\overline{\gamma (X)} \left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) \kappa (\overline{w_{\gamma }},w_{\gamma })\right. \\&\qquad \left. +\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{\gamma }},\overline{w_{\gamma }}) \right) =0, \\ \kappa&\left( \mathrm {ad}_{w_{\gamma }^r}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}} \right] \left( \mathrm {ad}_{w_{\gamma }^i}X\right) \right) \\&{\mathop {=}\limits ^{(4.24)}} \frac{1}{4\sqrt{-1}} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{\gamma })\right. \\&\qquad \left. -\gamma (X)\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X)) \right) }\kappa (w_{\gamma },\overline{w_{\gamma }})\right. \nonumber \\&\qquad \left. +\overline{\gamma (X)}\left( \gamma (X) \frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) \kappa (\overline{w_{\gamma }},w_{\gamma })\right. \\&\qquad \left. -\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{\gamma }},\overline{w_{\gamma }}) \right) =0, \\ \kappa&\left( \mathrm {ad}_{w_{\gamma }^r}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}} \right] \left( \mathrm {ad}_{w_{-\gamma }^r}X\right) \right) \\&{\mathop {=}\limits ^{(4.23)}} -\frac{1}{4} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{-\gamma }\right. )\\&\left. \qquad +\gamma (X)\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X)) \right) }\kappa (w_{\gamma },\overline{w_{-\gamma }})\right. \\&\qquad \left. +\overline{\gamma (X)}\left( \gamma (X) \frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) \kappa (\overline{w_{\gamma }},w_{-\gamma })\right. \\&\qquad \left. +\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{\gamma }},\overline{w_{-\gamma }}) \right) \\&\;\;\,=-\frac{1}{4} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{-\gamma })\right. \\&\qquad \left. +\overline{\gamma (X)} \overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)} -\gamma (D(X))\right) }\kappa (\overline{w_{\gamma }}, \overline{w_{-\gamma }}) \right) \\&\;\;\,=-\frac{1}{2}\mathrm {Re}\left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{-\gamma })\right) , \\ \kappa&\left( \mathrm {ad}_{w_{\gamma }^r}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}} \right] \left( \mathrm {ad}_{w_{-\gamma }^i}X\right) \right) \\&{\mathop {=}\limits ^{(4.24)}} -\frac{1}{4\sqrt{-1}} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{-\gamma })\right. \\&\qquad \left. -\gamma (X)\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X)) \right) }\kappa (w_{\gamma },\overline{w_{-\gamma }})\right. \nonumber \\&\qquad \left. +\overline{\gamma (X)}\left( \gamma (X) \frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) \kappa (\overline{w_{\gamma }},w_{-\gamma })\right. \\&\qquad \left. -\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{\gamma }},\overline{w_{-\gamma }}) \right) \\&\;\;\,=-\frac{1}{4\sqrt{-1}} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{-\gamma })\right. \\&\qquad \left. -\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{\gamma }},\overline{w_{-\gamma }}) \right) \\&\;\;\,=-\frac{1}{2}\mathrm {Im}\left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{-\gamma })\right) , \\ \end{aligned}$$
$$\begin{aligned} \kappa&\left( \mathrm {ad}_{w_{\gamma }^i}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}} \right] \left( \mathrm {ad}_{w_{\gamma }^i}X\right) \right) \\&{\mathop {=}\limits ^{(4.25)}} -\frac{1}{4} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{\gamma })\right. \\&\qquad \left. -\gamma (X)\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X)) \right) }\kappa (w_{\gamma },\overline{w_{\gamma }})\right. \nonumber \\&\qquad \left. -\overline{\gamma (X)}\left( \gamma (X) \frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) \kappa (\overline{w_{\gamma }},w_{\gamma })\right. \\&\qquad \left. +\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{\gamma }},\overline{w_{\gamma }}) \right) =0, \\ \kappa&\left( \mathrm {ad}_{w_{\gamma }^i}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}} \right] \left( \mathrm {ad}_{w_{-\gamma }^r}X\right) \right) \\&{\mathop {=}\limits ^{(4.24)}} -\frac{1}{4\sqrt{-1}} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{-\gamma })\right. \\&\qquad \left. +\gamma (X)\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X)) \right) }\kappa (w_{\gamma },\overline{w_{-\gamma }})\right. \nonumber \\&\qquad \left. -\overline{\gamma (X)}\left( \gamma (X) \frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) \kappa (\overline{w_{\gamma }},w_{-\gamma })\right. \\&\qquad \left. -\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{\gamma }},\overline{w_{-\gamma }}) \right) \\&\;\;\,=-\frac{1}{4\sqrt{-1}} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{-\gamma })\right. \\&\qquad \left. -\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{\gamma }},\overline{w_{-\gamma }}) \right) \\&\;\;\,=-\frac{1}{2}\mathrm {Im}\left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{-\gamma })\right) , \\ \kappa&\left( \mathrm {ad}_{w_{\gamma }^i}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}} \right] \left( \mathrm {ad}_{w_{-\gamma }^i}X\right) \right) \\&{\mathop {=}\limits ^{(4.25)}} \frac{1}{4} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{-\gamma })\right. \\&\qquad \left. -\gamma (X)\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X)) \right) }\kappa (w_{\gamma },\overline{w_{-\gamma }})\right. \nonumber \\&\qquad \left. -\overline{\gamma (X)}\left( \gamma (X) \frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) \kappa (\overline{w_{\gamma }},w_{-\gamma })\right. \\&\qquad \left. +\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{\gamma }},\overline{w_{-\gamma }}) \right) \\&\;\;\,=\frac{1}{4} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{-\gamma })\right. \\&\qquad \left. +\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{\gamma }},\overline{w_{-\gamma }}) \right) \\&\;\;\,=\frac{1}{2}\mathrm {Re}\left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{\gamma },w_{-\gamma }) \right) , \\ \kappa&\left( \mathrm {ad}_{w_{-\gamma }^r}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}} \right] \left( \mathrm {ad}_{w_{-\gamma }^r}X\right) \right) \\&{\mathop {=}\limits ^{(4.23)}} \frac{1}{4} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{-\gamma },w_{-\gamma })\right. \\&\qquad \left. +\gamma (X)\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X)) \right) }\kappa (w_{-\gamma },\overline{w_{-\gamma }})\right. \\&\qquad \left. +\overline{\gamma (X)}\left( \gamma (X) \frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) \kappa (\overline{w_{-\gamma }},w_{-\gamma })\right. \\&\qquad \left. +\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{-\gamma }},\overline{w_{-\gamma }}) \right) =0,\\ \kappa&\left( \mathrm {ad}_{w_{-\gamma }^r}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}} \right] \left( \mathrm {ad}_{w_{-\gamma }^i}X\right) \right) \\&{\mathop {=}\limits ^{(4.24)}} \frac{1}{4\sqrt{-1}} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{-\gamma },w_{-\gamma })\right. \end{aligned}$$
$$\begin{aligned}&\qquad \left. -\gamma (X)\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X)) \right) }\kappa (w_{-\gamma },\overline{w_{-\gamma }})\right. \nonumber \\&\qquad \left. +\overline{\gamma (X)}\left( \gamma (X) \frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) \kappa (\overline{w_{-\gamma }},w_{-\gamma })\right. \\&\qquad \left. -\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{-\gamma }},\overline{w_{-\gamma }}) \right) =0, \\ \kappa&\left( \mathrm {ad}_{w_{-\gamma }^i}X, \mathrm {Hess}_X\left[ H|_{{\mathcal {O}}} \right] \left( \mathrm {ad}_{w_{-\gamma }^i}X\right) \right) \\&{\mathop {=}\limits ^{(4.25)}} -\frac{1}{4} \left( \gamma (X)\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X))\right) \kappa (w_{-\gamma },w_{-\gamma })\right. \\&\qquad \left. -\gamma (X)\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}-\gamma (D(X)) \right) }\kappa (w_{-\gamma },\overline{w_{-\gamma }})\right. \nonumber \\&\qquad \left. -\overline{\gamma (X)}\left( \gamma (X) \frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) \kappa (\overline{w_{-\gamma }},w_{-\gamma })\right. \\&\left. \qquad +\overline{\gamma (X)}\overline{\left( \gamma (X)\frac{\gamma (b)}{\gamma (a)}- \gamma (D(X))\right) }\kappa (\overline{w_{-\gamma }},\overline{w_{-\gamma }}) \right) =0. \end{aligned}$$

Here, we used the consequences of Lemma 2.4: \(w_{\gamma }\in {\mathfrak {g}}_{\gamma }^{{\mathbb {C}}}\), \(\overline{w_{\gamma }}\in {\mathfrak {g}}_{{\overline{\gamma }}}^{{\mathbb {C}}}\), \({\mathfrak {g}}_{\gamma }\perp _{\kappa }{\mathfrak {g}}_{\gamma }\), \({\mathfrak {g}}_{\pm \gamma }\perp _{\kappa } {\mathfrak {g}}_{\pm {\overline{\gamma }}}\), where all different signs can be taken.

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Ratiu, T.S., Tarama, D. Geodesic flows on real forms of complex semi-simple Lie groups of rigid body type. Res Math Sci 7, 32 (2020). https://doi.org/10.1007/s40687-020-00227-2

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Keywords

  • Geodesic flow
  • Real semi-simple Lie group
  • Free rigid body
  • Cartan subalgebra
  • Bi-Hamiltonian structure
  • Integrable system
  • Equilibrium
  • Williamson type
  • Lyapunov stability

Mathematics Subject Classification:

  • 34D20
  • 53D25
  • 70E15
  • 70E45