Vol. 32, No. 2, 1970

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Jordan algebras and exceptional subalgebras of the exceptional algebra E6

Harry P. Allen and Joseph Cooley Ferrar

Vol. 32 (1970), No. 2, 283–297

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.

Mathematical Subject Classification
Primary: 17.10
Received: 14 October 1968
Published: 1 February 1970
Harry P. Allen
Joseph Cooley Ferrar