The close relationship which
exists between exceptional central simple Lie algebras, Cayley algebras, and
exceptional central simple Jordan algebras has been known for some time. The
representational point of view which the latter nonassociative algebras afford has led
to the complete classification of the Lie algebras G2 and F4, partial classification of
the Lie algebras D4 and E6, and to concrete realizations for forms of the above
algebras and the algebras E7 and E8. In the present paper we shall establish a
“coordinatization” theorem (Theorem 2) for exceptional simple subalgebras of the Lie
algebra L(J) of type E6, over an algebraically closed field of characteristic 0, in
terms of the annihilated subspace. We use this to give a new proof of the
well known conjugacy (see Dynkins Table 25) of split subalgebras of type
G2 or D4 or F4, of a split algebas of type D4 or F4 or E6 over a field of
characteristic 0 (Theorem 3). This is then applied to obtain new results in the
classification of D4 and E6 which are subsequently used in generalizing the above
conjugacy and extension of automorphism theorems to the (possibly) nonsplit
case.
