Triality algebra of a composition algebra

For a real composition algebra $A$ with its norm bilinear form, its triality algebra is

\operatorname{tri}(A)= \{(T_1,T_2,T_3)\in\mathfrak{so}(A)^3: T_1(xy)=T_2(x)y+xT_3(y)\text{ for all }x,y\in A\},

with componentwise commutator. Here $\mathfrak{so}(A)$ means the skew endomorphisms for the specified bilinear form, which may be indefinite for split $A$ . This fixes the order of the three slots.

Division inputs

For the positive-definite real composition algebras,

\begin{aligned} \operatorname{tri}(\mathbb R)&=0,& \operatorname{tri}(\mathbb C)&\cong\mathbb R^2\text{ with zero bracket},\\ \operatorname{tri}(\mathbb H)&\cong\mathfrak{sp}(1)^{\oplus3},& \operatorname{tri}(\mathbb O)&\cong\mathfrak{so}(8). \end{aligned}

Their dimensions are $0,2,9,28$ . The quaternionic output is the triple direct sum, while the octonionic output gives infinitesimal triality.

Derivations inside triality

A derivation $D$ gives the diagonal triple $(D,D,D)$ . Expanding two applications of the defining identity shows that componentwise commutators satisfy it again; thus the displayed set is a Lie subalgebra of $\mathfrak{so}(A)^3$ .

References

C. H. Barton and A. Sudbery, “Magic squares and matrix models of Lie algebras,” §4.1, Definition 2, Lemma 4.1 and the table on p. 13; our last two slots are interchanged relative to Definition 2. Paper.