On Jordan structure of nilpotent matrices from Lie algebra $\mathfrak{so}(N,\mathbb{C})$

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.