Centralisers of semi-simple elements are semidirect products

Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraically closed field, and let $s\in \mathbf{G}$ be a semisimple element. We show that the centraliser of $s$ is the semi-direct product of its identity component by its group of components. We then look at the case where $\mathbf{G}$ is defined over an algebraic closure of a finite field ${\mathbb{F}}_q$, and $F$ is a Frobenius endomorphism attached to an ${\mathbb{F}}_q$-structure on $\mathbf{G}$. We show that if the centraliser of $s$ is $F$-stable we have a semi-direct product decomposition of the $F$-fixed points.