Invariants of the adjoint action on nil-radicals of parabolic subalgebras in $B_n$, $C_n$, $D_n$

We consider the conjugation action of the unitriangular subgroup $N$ of one of the following groups $\mathrm{Sp}_{2n}$, $\mathrm O_{2n}$, $\mathrm O_{2n+1}$, on the nilradical of a parabolic subalgebra in the corresponding Lie algebra. We introduce the notion of an extended base in the set of positive roots. To each root of the extended base there corresponds an invariant with respect to the adjoint action of $N$. We show that these invariants are algebraically independent. Also, we estimate trancendence degrees of these invariants. Bibl. 6 titles.