On simple reducible Lie algebras of depth two

Abstract

We show that under certain circumstances simple finite-dimensional reducible graded Lie algebras of the form L − 2 ⊕ L − 1 ⊕ L 0 ⊕ L 1 ⊕ ⋯ ⊕ L k {L_{ - 2}} \oplus {L_{ - 1}} \oplus {L_0} \oplus {L_1} \oplus \cdots \oplus {L_k} can be given irreducible transitive gradations of the form M − 1 ⊕ M 0 ⊕ ⋯ ⊕ M [ k / 2 ] {M_{ - 1}} \oplus {M_0} \oplus \cdots \oplus {M_{[k/2]}} .