Let $G$ be a Lie group, and $H < G$ be isomorphic to $\operatorname{SL}(2, \mathbb{R})$. Now, each $h \in H$ acts on $G$ by conjugation $\phi_g : h \mapsto ghg^{-1}$, and this induces the adjoint representation $\operatorname{Ad}_h$ on $\mathfrak{g}$.
Now, any representation of $\operatorname{SL}(2, \mathbb{R})$ can be written as a the symmetric tensor product $\operatorname{Sym}^n(\mathbb{R}^2)$. Any element of this space can be written as a homogeneous polynomial $$p(A, B) = \sum_{i + j = n} c_i A^i B^j,$$ where $A = \pmatrix{1 \\ 0}$, and $B = \pmatrix{0 \\ 1}$. Left multiplication by elements of $H$ act on $p(A, B)$ through applying their $SL(2, R)$ counterpart as follows $h \cdot p(A, B) = p(h \cdot A, h \cdot B)$.
Now, what about the adjoint operator? How does it look like in terms of this irreducible representation?