A simple diagonal action

The socle of the acting group is $T\times T$, a direct product of two isomorphic nonabelian simple factors, and the point stabilizer is the diagonal subgroup $\mathrm{\Delta }(T)$ by construction. This stabilizer is maximal: if $\mathrm{\Delta }(T)<L\le T\times T$, choose $(a,b)\in L\setminus \mathrm{\Delta }(T)$. Multiplying by $(a^{-1},a^{-1})$ gives $(1,b{a}^{-1})\in L$ with $b{a}^{-1}\mathrm{\ne }1$. Conjugation by $\mathrm{\Delta }(T)$ and simplicity of $T$ then give $1\times T\le L$, and $\mathrm{\Delta }(T)(1\times T)=T\times T$. Thus $L=T\times T$.

A coset action is primitive exactly when its stabilizer is maximal, so step 1.1 makes this action primitive. Its socle and stabilizer are then exactly the defining features in [L1], and the action is a simple diagonal action.