On the compact real forms of the lie algebras of type E 6 and F 4

Abstract

We give a construction of the compact real form of the Lie algebra of type E6, using the finite irreducible subgroup of shape 33+3: SL3(3), which is isomorphic to a maximal subgroup of the orthogonal group Ω7(3). In particular we show that the algebra is uniquely determined by this subgroup. Conversely, we prove from first principles that the algebra satisfies the Jacobi identity, and thus give an elementary proof of existence of a Lie algebra of type E6. The compact real form of F4 is exhibited as a subalgebra.