Let $M$ be a spin-manifold of dimension $7$. Let $S\rightarrow M$ be a spin-bundle on $M$. Then Clifford multiplication ($c$) gives us the following isomorphism:
\begin{align*} c:i\Lambda^2\oplus\Lambda^4\oplus i\Lambda^6\rightarrow i\mathfrak{su}(S) \end{align*}
I want to understand what is the induced splitting of this isomorphism on the right hand side. How does one differentiate an imaginary two-form from a four-form as elements of trace-free Hermitian endomorphisms of $S?$
