Let $G$ be a Linear Algebraic Group (over algebraically closed field). We know that the connected component $G^o$ is a normal subgroup of finite index in $G$. Let $g$ be an element of $G$ which is not in $G^o$.

I want to understand Jordan decomposition of $g$ as a product of semisimple and unipotent elements. It seems to me that in characteristic 0 we have $g$ semisimple. However in char p it can be unipotent of certain kind. I want to know if my understanding is correct and if there is some neat classification.

Thanks a lot.

Let $G$ be a Linear Algebraic Group (over algebraically closed field). We know that the connected component $G^o$ is a normal subgroup of finite index in $G$. Let $g$ be an element of $G$ which is not in $G^o$.

I want to understand Jordan decomposition of $g$ as a product of semisimple and unipotent elements.

Modification from earlier question:

Can I choose representatives for $G/G^o$ consisting of semisimple elements alone? Of course the answer is 'no' in char p. But how bad is the scene.

Thanks a lot.