= Solution
An element $g\in G(k)$ is <semisimple element of an affine algebraic group>[semisimple] when its image in a faithful finite-dimensional representation is diagonalizable, and <unipotent element of an affine algebraic group>[unipotent] when that image has every eigenvalue equal to one. These conditions are independent of the faithful representation. The <Jordan decomposition in an affine algebraic group> is the unique factorization
$$
g=g_sg_u=g_ug_s
$$
with $g_s$ semisimple and $g_u$ unipotent.
In $G=\operatorname{GL}_2$, matrices with two distinct eigenvalues form a dense open subset and are semisimple, so semisimple elements are dense. Their full locus is not open: a scalar matrix is semisimple, but every neighbourhood of it contains a nontrivial Jordan block. The group also has nonidentity unipotent matrices.
For a contrasting example take $G=\mathbb G_a\times\mathbb G_m$. Its semisimple locus is $\{0\}\times\mathbb G_m$, which is closed and not dense, while $(a,1)$ for $a\ne0$ is unipotent and $(0,t)$ for $t\ne1$ is semisimple.
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