Derivations of nilpotent Lie algebras

Abstract

J. DIXMIER AND W. G. LISTER In a recent note Jacobson proved [l] that, over a field of characteristic 0, a Lie algebra with a nonsingular derivation is nilpotent. He also noted that the validity of the converse was an open question. The purpose of this note is to supply a strongly negative answer to that question and to point out some of the immediate problems which this answer raises. Suppose then that 4> is a field of characteristic 0 and that 2 is the 8 dimensional algebra over described in terms of a basis ei, e2, • ■ • , t?s by the following multiplication table: (1) [eu e2] = eB, (6) [e2, e4] = e6, (2) [eu e8] = e6, (7) [e2, e6] = — e7, (3) [eu e4] = e7, (8) [e3, e4] = — e6, (4) [eu et] = e8, (9) [e3, e6] = e7, (5) [eit e3] = e8, (10) [e4, e6] = — es. In addition [e 4. It is convenient to use a symmetry in the table above. Denote by A the linear transformation induced in 2 by the mapping