Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\phi_1:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{g}$ and $\phi_{2}:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{g}$ be complex Lie algebra morphisms. By composing $\phi_1$ and $\phi_2$ with the adjoint representation of $\mathfrak{g}$, we obtain two representations of $\mathfrak{sl}_2(\mathbb{C})$ on $\mathfrak{g}$. My question is then the following.

**Question:** If these two $\mathfrak{sl}_2(\mathbb{C})$-representations are isomorphic, does it follow that $\phi_1$ and $\phi_2$ are related by an element of the Lie algebra automorphism group $\text{Aut}(\mathfrak{g})$?

To provide some context, suppose that $\mathfrak{g}=\mathfrak{so}_{4n}(\mathbb{C})$. If $\lambda$ is a partition of $4n$ having only even parts with each part appearing an even number of times, then $\lambda$ corresponds to exactly two distinct nilpotent orbits, $\mathcal{O}_1$ and $\mathcal{O}_2$, in $\mathfrak{so}_{4n}(\mathbb{C})$. Now, let $\phi_1:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{so}_{4n}(\mathbb{C})$ and $\phi_2:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{so}_{4n}(\mathbb{C})$ be Lie algebra maps satisfying $\phi_1(e)\in\mathcal{O}_1$ and $\phi_2(e)\in\mathcal{O}_2$. As discussed above, each map gives a representation of $\mathfrak{sl}_2(\mathbb{C})$ on $\mathfrak{so}_{4n}(\mathbb{C})$. These two representations are actually isomorphic. While $\phi_1$ and $\phi_2$ cannot be related by an inner automorphism of $\mathfrak{so}_{4n}(\mathbb{C})$ (as $\mathcal{O}_1\neq\mathcal{O}_2$), they are nevertheless related by a Lie algebra automorphism.