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

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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. 
@article{ZNSL_2023_528_a4,
     author = {M. V. Babich},
     title = {On {Jordan} structure of nilpotent matrices from {Lie} algebra $\mathfrak{so}(N,\mathbb{C})$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {79--90},
     publisher = {mathdoc},
     volume = {528},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2023_528_a4/}
}
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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/