The triple direct sum of sp(1)

The real Lie algebra $\mathfrak{sp}(1)^{\oplus3}$ is the direct sum of three copies of $\mathfrak{sp}(1)$ . For triples, its bracket is

[(X_1,X_2,X_3),(Y_1,Y_2,Y_3)] =([X_1,Y_1],[X_2,Y_2],[X_3,Y_3]).

Its real dimension is $9$ , and its three simple factors commute.

Quaternionic triality

There is a real Lie algebra isomorphism

\operatorname{tri}(\mathbb H)\cong\mathfrak{sp}(1)^{\oplus3}.

This records the full triality algebra; quaternion derivations alone form only a three-dimensional algebra.

References

C. H. Barton and A. Sudbery, “Magic squares and matrix models of Lie algebras,” §4.1, the derivation and triality table on p. 13. Paper.