Octonion conjugation, norm, and inner product

Every octonion $x\in\mathbb O$ splits uniquely as $x=\operatorname{Re}(x)+\operatorname{Im}(x)$ , with real and imaginary parts. Its conjugate, norm, and inner product are

x^*=\operatorname{Re}(x)-\operatorname{Im}(x), \qquad N(x)=xx^*=x^*x=\lVert x\rVert^2, \qquad \langle x,y\rangle=\operatorname{Re}(xy^*).

The last formula is the standard Euclidean inner product on the eight-dimensional real vector space $\mathbb O$ .

Identities

Conjugation is a real-linear involutive antiautomorphism:

(x^*)^*=x, \qquad (xy)^*=y^*x^*.

The quadratic norm composes,

N(xy)=N(x)N(y),

and hence $\lVert xy\rVert=\lVert x\rVert\lVert y\rVert$ . Every nonzero octonion has the two-sided inverse

x^{-1}=\frac{x^*}{N(x)}.

Polarization and conventions

One also encounters

\langle x,y\rangle =\frac12\bigl(N(x+y)-N(x)-N(y)\bigr) =\frac12(xy^*+yx^*).

The final expression is real, so it is identified with a scalar multiple of the unit. Some composition-algebra texts define the polar form without the factor $1/2$ ; their trace bilinear form is therefore twice the Euclidean inner product.

Orthogonal splittings

The imaginary octonions are the orthogonal complement of $1$ , giving

\mathbb O=\mathbb R1\oplus\operatorname{Im}(\mathbb O).

After choosing a unit imaginary octonion $i$ , the copy $\mathbb C=\operatorname{span}_{\mathbb R}\{1,i\}$ has a six-dimensional orthogonal complement, yielding the model $\mathbb O\cong\mathbb C\oplus\mathbb C^3$ .

References

John C. Baez, “The Octonions,” Bulletin of the American Mathematical Society 39 (2002), 145–205. DOI record. Relevant: §§2.1–2.2.
John H. Conway and Derek A. Smith, On Quaternions and Octonions, A K Peters, 2003. DOI record. Relevant: Chapters 3–4.