On the density of semisimple matrices in indefinite scalar product spaces

For an indefinite scalar product $[x,y]_B = x^HBy$ for $B= \pm B^H \in \mathbf{Gl}_n(\mathbb{C})$ on $\mathbb{C}^n \times \mathbb{C}^n$, it is shown that the set of diagonalizable matrices is dense in the set of all $B$-normal matrices. The analogous statement is also proven for the sets of $B$-selfadjoint, $B$-skewadjoint and $B$-unitary matrices.