Complexification of H

The complexified H algebra is the tensor product

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

with complex-linear extension of the product on H. Its unit is $1\otimes1$ , and its complex dimension is $4$ .

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 $M_2(\mathbb C)$ . One explicit realization sends the quaternion generators to $i\mapsto\operatorname{diag}(i,-i)$ and $j\mapsto\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$ ; the four basis images span $M_2(\mathbb C)$ .

This complexification has real dimension $8$ . 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.