Automorphisms of nil-triangular subrings in Chevalley algebra of orthogonal type

Any Chevalley algebra over an associative commutative ring K with the identity is characterized by Chevalley base that correspondents to each indecomposable root system Ф. All elements er (r Ф+ ) of Chevalley base give a base of subalgebra NФ(K) which is said to be nil-triangular. Automorphisms of algebras NФ(K) were described by Y. Cao, D. Jiang and D. Wang (J. Algebra, 2007) at K = 2K for Lie type Bn, Cn or F4 and under similar restrictions for other types. Their description uses only non-standard Gibbs’s automorphisms; in our terminology it is a hypercentral automorphisms of height 2 or 3 (for type Cn). Our main purpose is to describe the automorphism group A of the Lie ring NФ(K). The algebra NФ(K) of Lie type An-1 can be represented as Lie algebra which associated to the algebra NT(n, K) of all nil-triangular matrices over K. The automorphism group of the ring NT(n, K) and of its associated Lie ring (i. e., A for the type An) described earlier V. M. Levchuk (1983). A. V. Litavrin has described the automorphism group A for Lie type Cn recently. In the present paper we find non-standard automorphisms of the algebra NФ(K) for orthogonal types, when the condition K = 2K isn’t satisfied. It seems that if annihilator of element 2 in K is non-zero, then the largest height of hypercentral automorphisms grows together with the Lie rank. Also, we find automorphisms of the algebra NФ(K) of type Dn which are non-standard module second member of lower central series and generate the subgroup of A that isomorphic to certain subgroup S in SL(2, K); in particularly, S = SL(2, K) at 2K = 0. The standard automorphisms together with constructed non-standard automorphisms generate every automorphisms of the algebra NФ(K). For all classical types of rank > 4 our results show that the automorphism group A is the product of subgroups of the central and induced ring automorphisms and the automorphism group of the algebra NФ(K). We use developed earlier methods, in particularly, a special representation of the algebras NФ(K) of classical types. The results can be used in development of cryptographic methods.