Complexification of C

The complexified C algebra is the tensor product

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

with complex-linear extension of the product on C. Its unit is $1\otimes1$ , and its complex dimension is $2$ .

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 isomorphic to $\mathbb C\times\mathbb C$ , so its complex dimension is two, not one. In a presentation $\mathbb C[u]/(u^2+1)$ , evaluation at $u=\pm i$ gives the two factors.

This complexification has real dimension $4$ . 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.