Let $k$ be a field, and let $G$ be a connected reductive group over $k$. Form the adjoint quotient $G \rightarrow G^{\mathrm{ad}}$, which is the quotient of $G$ by its center.
Is the map on $k$-rational points $G(k) \rightarrow G^{\mathrm{ad}}(k)$ surjective? If not, are there some conditions when this is true?
It is definitely true if $k$ is algebraically closed.
I am particularly interested in the case where $k = \mathbb{R}$. A similar statement seems to be claimed in the proof of Proposition 5.7(a) in Milne's notes on Shimura varieties (https://www.jmilne.org/math/xnotes/svi.pdf). Is there some Galois cohomology argument here?