Maybe it is not serious, but then in a sense none of the examples in this thread are.

Jordan decomposition represents every operator on a finite-dimensional space in a unique way as $S+N$ where $S$ is a diagonalizable operator, $N$ is a nilpotent operator, and $SN=NS$.

The Jordan decomposition of the zero operator is $0+0$. It thus is the only operator which is diagonalizable and nilpotent at the same time.

Similarly, the zero Lie algebra is the only one which is both semisimple and nilpotent. And one might also argue that it is *not* simple. Or is it?...