TITLE: Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$
QUESTION [5 upvotes]: 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 $v$, $\Delta_{\pm}$

REPLY [2 votes]: 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 $\bigl( \begin{smallmatrix} & -I \\ I &  \end{smallmatrix} \bigr)$.