gl(1,H) — quaternionic general linear Lie algebra

The quaternionic general linear Lie algebra $\mathfrak{gl}(1,\mathbb H)$ is the real vector space

\mathfrak{gl}(1,\mathbb H)=M_{1}(\mathbb H),\qquad [X,Y]=XY-YX.

Trace and scalar conventions

Quaternionic matrix multiplication is associative, so its commutator is a real Lie bracket. Each quaternionic entry has four real coordinates, giving the displayed dimension.

Quaternionic entries do not make this a Lie algebra over a commutative field $\mathbb H$ . All scalar-linear categories in this record are over $\mathbb R$ .