I am currently trying to understand the notion of a multiplicative group scheme.
By Milne (https://www.jmilne.org/math/CourseNotes/AGS.pdf, 5.11) it is a group scheme that becomes diagonalizable over a field extension. In (https://ncatlab.org/nlab/show/multiplicative+group+scheme) this is denoted as $G\otimes_k K$ is diagonalizable.
Now for my question: How exactly is $G\otimes_k K$ defined? In my opinion, it should be the base change of the group scheme, i.e. $G \times_k K \rightarrow K$.