Let $G$ be an algebraic group, and $\mu$ is the multiplication in $G$. Define a morphism $G \times G \times G \times G \rightarrow G \times G$ by $\mu \times \mu$, and let $X$ be the inverse image of the diagonal $\{(x,x)|x \in G\}$. If $G$ is connected, prove that $X$ is a closed irreducible subset of $G \times G \times G \times G$.
The closedness part of this proof is clear. But I have no idea as to the connectedness or the irreducibility of $X$.
A morphism between two varieties is continuous, and the multiplication, as well as inverse map of an algebraic group are all morphisms. These can be applied to determine whether a set is closed. But, what are the efficient ways to prove connectedness?
Many thanks~