Let $\mathfrak{g}$ be the Lie algebra of a matrix Lie group. Furthermore, let us consider the following $\mathfrak{g}$-valued $p$-form and $\mathfrak{g}$-valued $q$-form: \begin{equation} \begin{array}{cc} \zeta = \zeta^\alpha \otimes T_\alpha \; ,& \eta = \eta^\alpha \otimes T_\alpha \end{array} \end{equation} where $\zeta^\alpha \in \Omega^p(P)$, $\eta^\alpha \in \Omega^q(P)$ and $\{ T_\alpha \}$ is a basis of $\mathfrak{g}$. Then I'm trying to show that: \begin{equation} \zeta \wedge \eta = T_\alpha T_\beta \zeta^\alpha \wedge \eta^\beta \tag{1} \end{equation} Can someone help me or give me a nudge in the right direction? The only possibility I see is using the definition: \begin{equation} (\zeta \wedge \eta)(X_1,\ldots,X_{p+q}) = \frac{1}{p!q!} \sum\limits_{P \in S_r} \mathrm{sgn}(P) \zeta(X_{P(1)},\ldots,X_{P(p)}) \eta (X_{P(p+1)},\ldots,X_{P(p+q)}) \end{equation} but is don't really understand how to manipulate this in order to get equation $(1)$.
I'm not sure if the notation used above is conventional notation for mathematicians, so please let me know if I need to clarify anything.