Complexification of O

The complexified O algebra is the tensor product

\mathbb O_{\mathbb C}=\mathbb O\otimes_{\mathbb R}\mathbb C, \qquad (a\otimes z)(b\otimes w)=(ab)\otimes zw,

with complex-linear extension of the product on O. Its unit is $1\otimes1$ , and its complex dimension is $8$ .

Coefficient involution

The composition-algebra conjugation is $a\otimes z\mapsto\overline a\otimes z$ . It is complex-linear: it does not conjugate the new scalar $z$ . The composition norm is the complex quadratic extension of the real norm, not a positive Hermitian norm.

Split description

It is the split octonion algebra over $\mathbb C$ , and remains alternative and nonassociative. A nondegenerate quadratic form in eight variables over $\mathbb C$ is isotropic, so this algebra is not a division algebra.

This complexification has real dimension $16$ . It has a different catalogue identity from the original real algebra, and forgetting complex scalar multiplication is a separate category view.

References

Alberto Elduque, Composition algebras, Section 2.1, equation (7); Theorem 2.11 and Corollary 2.12.