In the first answer to the [linked question][1] 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 types" is mentioned but there is nothing there about it as far as I can find. 


  [1]: https://mathoverflow.net/questions/148833/chevalley-groups-over-an-arbitrary-ring