Products of similar matrices

Let $A$ and $B$ be $n \times n$ matrices of determinant $1$ over a field $K$, with $n >2$ or $|K|>3$. We show that if $A$ is not a scalar matrix, then $B$ is a product of matrices similar to $A$. Analogously, we conjecture that if $a$ and $b$ are elements of a semisimple algebraic group $G$ over a field of characteristic zero, and if there is no normal subgroup of $G$ containing $a$ but not $b$, then $b$ is a product of conjugates of $a$. The conjecture is verified for orthogonal groups and symplectic groups, and for all semisimple groups over local fields. Thus, in a connected, semisimple Lie group with finite center, the only conjugation-invariant subsemigroups are the normal subgroups.