A question in spin geometry in dimension 8

Question

$\DeclareMathOperator\trace{trace}\DeclareMathOperator\End{End}\DeclareMathOperator\Trace{Trace}$This is to understand a very specific isomorphism in dimension $8$. In dimension $4$ for a spin$^c$ manifold $M^4$ and $S=S_+\oplus S_-\rightarrow M$ a spinor bundle, we have the following isomorphism of normed vector spaces given by the Clifford multiplication: \begin{align*} c:i\Omega^2_+\rightarrow i\mathfrak{su}(\Gamma(S_+)) \end{align*} the norm on the rhs being \begin{align*} \langle T,S\rangle:=\frac{1}{4}\text{trace}(TS^*). \end{align*} I wish to understand the similar isomorphism in dimension $8$. In dimension $8$ the following is an isomorphism of two normed vector spaces: \begin{align*} c:i\Omega^2\oplus\Omega^4_+\rightarrow i\mathfrak{su}(\Gamma(S_+)). \end{align*} $c$ denotes Clifford multiplication. The norm on the lhs is the usual norm on forms induced by the Riemannian metric and the norm on the rhs is the norm from $\End(\Gamma(S_+))$, i.e., for $T,S\in \End(\Gamma(S_+))$, \begin{align*} \langle T,S\rangle:= a\trace(TS^*). \end{align*} For $T\in i\mathfrak{su}(\Gamma(S_+))$, $T=T^*$. Hence, \begin{align*} |T|^2=a\trace(T^2). \end{align*} We need to figure out what is $a$. Let's check it in the Kähler case. $S_+=\Omega^0\oplus\Omega^{0,2}\oplus\Omega^{0,4}$, \begin{align*} c(i\omega)=\begin{bmatrix} 4&0&0\\ 0&0&0\\ 0&0&-4 \end{bmatrix}, c(\omega^2)=\begin{bmatrix} -12&0&0\\ 0&4&0\\ 0&0&-12 \end{bmatrix} \end{align*} \begin{align*} c(i\omega)^2=\begin{bmatrix} 16&0&0\\ 0&0&0\\ 0&0&16 \end{bmatrix}, c(\omega^2)^2=\begin{bmatrix} 144&0&0\\ 0&16&0\\ 0&0&144 \end{bmatrix}\\ \frac{1}{8}\Trace\bigl(c(i\omega)^2\bigr)=\frac{32}{8}=4=|\omega|^2\\ \frac{1}{16}\Trace\bigl(c(\omega^2)^2\bigr)=\frac{144\times 2+6\times 16}{16}=24=|\omega^2|^2. \end{align*} It seems $a=\frac{1}{8}$ for $T\in c(i\Omega^2)$ and $a=\frac{1}{16}$ for $T\in c(\Omega^4_+)$. Now it seems a bit odd as I was hoping to get the same constant. Given $\phi\in\Gamma(S_+)$, $\phi^*\otimes\phi-\frac{1}{8}|\phi|^2\mathrm{Id}\in i\mathfrak{su}(\Gamma(S_+))$. How do we calculate the norm of this element?