Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$ I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over $\mathbb{R}$). More concretely, it would be very useful to know:


*

*The image of the basis elements $e_k$

*The image of the volume element $v=e_1e_2e_3e_4e_5e_6e_7e_8$

*The equations of the $\pm$-eigespaces of $v$, $\Delta_{\pm}$
 A: In order to establish isomorphism of Clifford $C_8$ with $M_{16}\mathbb R$ it is enough to define 8 letters generators which square to $-1$ and anticommute. Define
$e_k$=$$\begin{pmatrix} L_k & \\
& R_k \end{pmatrix}
$$
where $L_k$ and $R_k$ are left and right multiplication by imaginary base octonions, $k$=1..7. Last letter $e_8$=$$\begin{pmatrix} & C \\ -C & \end{pmatrix} $$. In this presentation we can see that first 7 letters generate $C_7$=$M_8\mathbb R$+ 
$M_8\mathbb R$. 
Alternatively you can use $\bigl( \begin{smallmatrix} L &  \\  & -L \end{smallmatrix} \bigr)$ and $\bigl( \begin{smallmatrix} & -I \\ I &  \end{smallmatrix} \bigr)$.
A: Here's a standard explicit formula:  Let $\mathbb{O}\simeq\mathbb{R}^8$ denote the algebra of octonions, and for $x\in\mathbb{O}$, let $L_x$ (respectively $R_x)$ denote the linear map from $\mathbb{O}$ to itself generated by left (respectively, right) multiplication by $x$ and let $C:\mathbb{O}\to\mathbb{O}$ be conjugation in the octonions.  Now define $\rho(x):\mathbb{O}\oplus\mathbb{O}\to\mathbb{O}\oplus\mathbb{O}\simeq\mathbb{R}^{16}$ 
to be the matrix in $\mathbb{R}(16)$
$$
\rho(x) = \begin{pmatrix} 0 & CR_x\\ - CL_x & 0\end{pmatrix}.
$$
Then one easily computes that $\rho(x)^2 = -|x|^2 \mathrm{Id}_{\mathbb{O}\oplus\mathbb{O}}$, so $\rho$ extends to a homomorphism $\rho:Cl(8)\to \mathbb{R}({16})$, which is necessarily an isomorphism.
I should also remark that the two eigenspaces of $v$ are the two given $\mathbb{O}$-summands in $\mathbb{O}\oplus\mathbb{O}$; which is $\Delta_+$ depends on which orientation of $\mathbb{O}$ you choose.
