Lie algebras of type 𝐹

Abstract

Chevalley and Schafer [4 ]2 have shown that the exceptional simple Lie algebra F4 of dimension 52 over an arbitrary algebraically closed field Q2 of characteristic 0 is the derivation algebra of the unique exceptional simple Jordan algebra of dimension 27 over Q2. In this paper we show that a Lie algebra 2 over an arbitrary field 1D of characteristic 0 is of type F if and only if 2 is isomorphic to the derivation algebra Z(3) of an exceptional central simple Jordan algebra a over (D. The proof given for this theorem requires a characterization of the automorphisms of ZQ() over R. We prove that every automorphism of Z(a) has the form D-+SDS-1 for a unique automorphism S of a. The classification of Lie algebras of type F over 4) is reduced to the problem of classifying exceptional central simple Jordan algebras over (D, since it is shown that Z ,1)Z)($a2) if and only if 2 In the last section of this paper the three exceptional central simple Jordan algebras over a real closed field are exhibited and their derivation algebras are the real closed Lie algebras of type F.