Lie-orthogonal operators

Lie-orthogonal operators on a finite-dimensional Lie algebra are studied. The components of the ascending central series and the radical are invariant with respect to any Lie-orthogonal operator. Only solvable Lie algebras with solvability degree not greater than two admit Lie-orthogonal without eigenvalues 1 and −1. Lie-orthogonal operators on a simple Lie algebra are trivial. This gives the description of Lie-orthogonal operators for semi-simple and reductive algebras. Lie-orthogonal operators of some classes of Lie algebras are directly computed. Thus, the equivalence classes of Lie-orthogonal operators on a Heisenberg algebra form the symplectic group of an appropriate dimension.