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 A≅A ⊗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.
