Uniqueness of Branching through regular unipotent elements

Let $\mathrm G$ be a complex simple algebraic group and let $\mathrm G_0\subset \mathrm G$ be a closed connected subgroup containing a regular unipotent element of $\mathrm G$, with semisimple rank at least $2$. Using Dynkin's classification, we prove that the restriction of an irreducible finite-dimensional representation of $\mathrm G$ to $\mathrm G_0$ determines the representation up to an outer automorphism of $\mathrm G$ preserving $\mathrm G_0$. We extend this method to the diagonal embedding $\mathrm G_0\hookrightarrow \mathrm G\times \mathrm G$ for the specific pairs $(\mathrm{SO}_{2k}(\mathbb C) \times\mathrm{SO}_{2k}(\mathbb C),\,\mathrm{SO}_{2k-1}(\mathbb C))$, $(E_6\times E_6,\,F_4)$ and $(Spin_8(\mathbb C) \times Spin_8(\mathbb C), G_2)$ and show that uniqueness continues to hold. Finally, we give examples showing that, in the diagonal setting, restriction to the principal $\mathrm{SL}_2(\mathbb C)$ alone is not sufficient to establish uniqueness.