Forgive me for potentially coming up with a new name for what I am about to describe. Let's say we have a scalar field $\phi^a$ which transforms with respect to the adjoint representation of some Lie algebra $G$. Is it always possible to transform this field such that only the $\phi^a$ corresponding to some Cartan subalgebra is nonzero?

If so, is it further possible to transform such that $\phi^a$ lies in only one direction?


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