Abstract

We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo H -type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type Bn with |2|-grading do not contain non-Heisenberg pseudo H -type Lie algebras as their negative nilpotent part, while the complex simple Lie algebras of types An, Cn and Dn provide such a possibility. Among exceptional algebras only F4 and E6 contain non-Heisenberg pseudo H-type Lie algebras as their negative part of |2|-grading. An analogous question addressed to real simple graded Lie algebras is more delicate, and we give results revealing the main differences with the complex situation.