In this paper we
show how to construct all central simple Lie algebras of type D4 over an
algebraic number field. The construction that we use is a special case of a
modified version of a construction due to G. B. Seligman. The starting point for
the construction is an 8-dimensional nonassociative algebra with involution
CD(ℬ,μ) that is obtained by the Cayley-Dickson doubling process from a
4-dimensional separable commutative associative algebra ℬ and a nonzero
scalar μ. The algebra CD(ℬ,μ) is used as the coefficient algebra for a Lie
algebra 𝒦(CD(ℬ,μ),γ) that can be roughly described as the Lie algebra
of 3 × 3-skew hermitian matrices with entries from CD(ℬ,μ) relative to
the involution X → γ−1Xtγ, where γ is an invertible diagonal matrix with
scalar entries. We show that any Lie algebra of type D4 over a number field
can be constructed as 𝒦(CD(ℬ,μ),γ) for some choice of ℬ, μ and γ. We
also give isomorphism conditions for two Lie algebras constructed in this
way.
