# Covariantly constant Lie algebra-valued field with Dirichlet boundary condition

I have a question about a statement in Witten's paper 'Analytic Continuation of Chern-Simons Theory' (https://arxiv.org/abs/1001.2933). On page 66, below equation 4.13, he discusses a Lie algebra-valued field $$\phi_t$$ which is covariantly constant with respect to a gauge field, $$A_{\mu}$$ on a 4-manifold $$I \times M$$ ($$I$$ is the interval), and in addition commutes with a number of other Lie algebra valued fields, i.e.,

$$D_{\mu}\phi_t=\partial_{\mu}\phi_t+[A_{\mu},\phi_t]=0,~~[\phi_{\mu},\phi_t]=0~~(\textrm{for } \mu\neq t).$$ Witten then claims that when the Dirichlet boundary condition $$\phi_t=0|_M$$ is obeyed, the unique solution to these equations is $$\phi_t=0$$

How does one show this?

It seems to me that $$[\phi_{\mu},\phi_t]=0$$ implies that $$\phi_t\in \frak{t},$$ where $$\frak{t}$$ is the commutative subalgebra of the Lie algebra. Then, the remaining equation becomes $$\partial_{\mu}\phi_t=0,$$ which implies that $$\phi_t$$ is a constant, and therefore $$\phi_t=0$$ everywhere due to the boundary condition. However, this does not seem like the most general way of solving these equations.