Vol. 24, No. 1, 1968

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Lie algebras of type D4 over algebraic number fields

Harry P. Allen

Vol. 24 (1968), No. 1, 1–5

If A is a nonassociative algebra over an algebraically closed field L, then the classification problem for A is the determination of all algebras A over Φ L where AA ΦL. This brief note studies this problem for the case where A is the Lie algebra D4 and Φ is a (finite) algebraic number field. The main result is a type of Hasse principle which tells us that a Lie algebra L (over Φ) of type D4 has known type if the algebra LΦp has known type for every completion Φp of Φ. This is used in §3 to obtain canonical splitting fields for Lie algebras of type D4 over Φ. Although the results are inconclusive with regard to the existence or nonexistence of new algebras, it indicates a (twisted) construction, which if nonvacuous, would yield new exceptional algebras of type D4III1.

Mathematical Subject Classification
Primary: 17.00
Received: 17 January 1967
Published: 1 January 1968
Harry P. Allen