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In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? Carter's book "Simple groups of Lie type" is mentioned but there is nothing there about it as far as I can find.

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    $\begingroup$ For $k$ algebraically closed it's a bijection (I guess, essentially by definition of quotient). Injectivity follows for arbitrary fields. $\endgroup$
    – YCor
    Commented Apr 10 at 21:48
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    $\begingroup$ To elaborate on YCor's comment: $1\to Z(k^{sep})\to G(k^{sep})\to G^{ad}(k^{sep})\to 1$ is an exact sequence, and taking $\mathrm{Gal}(k^{sep}/k)$-fixed points is a left exact functor. $\endgroup$
    – Kenta Suzuki
    Commented Apr 10 at 23:41

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