Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $Spec\mathbb{C}$, and the underlying set of $G$ is a complex algebraic variety.
By action of $G$ on $X$, I mean a morphism $\alpha:X\times_{\mathbb{C}}G\to X$ of $\mathbb{C}$-schemes such that: for any $\mathbb{C}$-scheme $T$, one has that $\alpha(T):X(T)\times G(T)\to X(T)$ is an action of the group $G(T)$ on set $X(T)$; by this I can write $\alpha(T)(x,g)=x\cdot g$ without problems.
My doubt is: if I take a couple of points $x\in X$ and $g\in G$, has $\alpha(x,g)$ sense? I think no!, becasue in general $|X\times_{\mathbb{C}}G|$ is not in bijection with $|X|\times|G|$, as detailed in Stacks Project.
In consequence, if $P$ is a principal $G$-bundle on $X$: how can I construct the adjoint bundle $Ad(P)$?
EDIT As usual in algebraic geometry: by princpal $G$-bundle $P$ on $X$ I mean a locally isotrival $G$-fibration $P$ on $X$, that is a $G$-fibration trival on an étale covering of $X$!