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 $v$, $\Delta\_{\pm}$



Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$
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.


