Geodesic
On Jordan structure of nilpotent matrices from Lie algebra $\mathfrak{so}(N,\mathbb{C})$
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXV, Tome 528 (2023), pp. 79-90 
M. V. Babich. On Jordan structure of nilpotent matrices from Lie algebra $\mathfrak{so}(N,\mathbb{C})$. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXV, Tome 528 (2023), pp. 79-90. http://geodesic.mathdoc.fr/item/ZNSL_2023_528_a4/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The Jordan structure of matrices of the Lie algebra of a complex orthogonal group, nilpotent case, is considered. These matrices have an arbitrarily complicated Jordan structure, under the known condition that the number of Jordan blocks of even size is even. A normal form for such matrices is proposed. Gram matrices of Jordan chains are described.

[1] F. R. Gantmakher, Teoriya matrits, Nauka, M., 1966

[2] I. M. Gelfand i M. I. Naimark, “Unitarnye predstavleniya klassicheskikh grupp”, Tr. matem. instituta im. V. I. Steklova, 36, 1950

[3] M. V. Babich, “On birational Darboux coordinates on coadjoint orbits of classical complex Lie groups”, Zap. nauchn. semin. POMI, 432, 2014, 36–57

[4] M. V. Babich, “Diagrammy Yunga i stratifikatsiya prostranstva kvadratnykh kompleksnykh matrits”, Zap. nauchn. semin. POMI, 433, 2015, 41–64

[5] M. V. Babich, “Birational Darboux coordinates on nilpotent coadjoint orbits classical complex Lie groups, Jordan blocks $2\times 2$”, Zap. nauchn. semin. POMI, 465, 2017, 5–12