When is $||xy\pm yx|| \le 1$?

Question

For $2 \times 2$ matrices $x,y$ when can we have $\| xy-yx \| \le 1$ and similarly when is $\| xy+yx \| \le 1$? How about in the special case when $x$ and $y$ are hermitian i.e., $x^\dagger = x$ and $y^\dagger = y~$? Here $\dagger = {\rm conjugate-transpose}$, and $\|M\| =\sqrt{ \operatorname{Tr}M^\dagger M}$.