# nLab split octonions

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

Up to isomorphism there is only one split composition algebra of dimension 8; this is called the algebra of split octonions.

## Definition

The multiplication may be deduced from the fundamental result on doubling multiplication or may be expressed as follows. Denote scalars by letters like $r, s$ and 3-vectors by letters like $x, y$. Let $\langle x, y \rangle$ denote the standard inner product

$x_1 y_1 + x_2 y_2 + x_3 y_3$

and let $x \wedge y$ denote the standard cross-product, so that $\langle x \wedge y, z \rangle = det(x, y, z)$. Elements of $V$ are represented by $2 \times 2$ arrays

\left( \begin{aligned} r & x\\ y & s \end{aligned} \right)

and multiplication is given by the following formula, highly reminiscent of matrix multiplication but with some cross-product cross terms:

\left( \begin{aligned} r & x\\ y & s \end{aligned} \right) \cdot \left( \begin{aligned} r' & x'\\ y' & s' \end{aligned} \right) = \left( \begin{aligned} r r' + \langle x, y' \rangle & r x' + s' x + y \wedge y'\\ r' y + s y' + x \wedge x' & \langle y, x' \rangle + s s' \end{aligned} \right)

The norm is given by a kind of determinant formula

N\left( \begin{aligned} r & x\\ y & s \end{aligned} \right) = r s - \langle x, y \rangle