Find a subgroup of $GL(2, \mathbb C)$ which does not contain the semisimple and unipotent parts of all its elements.
Attempt: According to the theory, there is no chance to find one counterexample in the family of closed subgroups. I can't find good not closed subgroups, because the one I try to think of are created via relations on eigenvalues (e.g all eigenvalues are integer powers of some trascendent $\alpha$). These kind of relations turn out to be preserved in the semisimple and unipotent parts.
Just consider the group generated by \begin{bmatrix} e & 1 \\ 0 & e \end{bmatrix}
None of its elements but $I_2$ are semisimple nor unipotent.