The Casimir element of a rational representation is an endomorphism of G-modules

Given: A semisimple algebraic group $G$ over a characteristic-zero field $k$ with $\mathfrak{g}=\mathrm{Lie}(G)$, a finite-dimensional rational representation $(V,r)$ with differential $\rho :\mathfrak{g}\to \mathfrak{g}\mathfrak{l}(V)$, and $\bar{\mathfrak{g}}=\rho (\mathfrak{g})\mathrm{\ne }0$.

Quotients of semisimple Lie algebras. $\mathfrak{g}$ is semisimple (The Lie algebra of a semisimple group in characteristic zero is semisimple), and every quotient of a finite-dimensional semisimple characteristic-zero Lie algebra is semisimple (Ideals and quotients of semisimple Lie algebras); hence $\bar{\mathfrak{g}}\cong \mathfrak{g}\mathrm{/}\ker \rho$ is semisimple.

Nondegenerate trace form. The representation $\bar{\mathfrak{g}}\hookrightarrow \mathfrak{g}\mathfrak{l}(V)$ is faithful and finite-dimensional, so its trace form $B_{\rho }(x,y)=\mathrm{tr}(xy)$ for $x,y\in \bar{\mathfrak{g}}$ is nondegenerate and invariant (Trace forms of faithful representations of semisimple Lie algebras are nondegenerate).

Casimir element. For a semisimple Lie algebra with nondegenerate invariant form $B$, a basis $(e_i)$ and the $B$-dual basis $(e_i^{\mathrm{\prime }})$, the element ${\mathrm{\Omega }}_{B_{\rho }}={\sum }_i{e}_i{e}_i^{\mathrm{\prime }}\in U(\bar{\mathfrak{g}})$ is independent of the basis. Its action $c_V={\sum }_i{e}_i\circ e_i^{\mathrm{\prime }}\in {\mathrm{End}}_k(V)$ is an endomorphism of $V$ as a $\bar{\mathfrak{g}}$-module, and $\mathrm{tr}(c_V{\mathrm{\mid }}_V)={\sum }_{i}B(e_i,e_i^{\mathrm{\prime }})=\dim \bar{\mathfrak{g}}$ (Casimir operator relative to an invariant form, The Casimir operator is basis-independent and intertwining).

Endomorphisms of $V$. The space ${\mathrm{End}}_k(V)\cong V^{\ast }{\otimes }_{k}V$ is a finite-dimensional rational representation of $G$ with $(g\cdot f)(v)=r(g)f(r(g{)}^{-1}v)$, whose fixed points are exactly the $G$-module endomorphisms of $V$ (Tensor products, exterior powers and Hom spaces of finite-dimensional rational representations are rational, Rational representations and comodules of an affine group scheme).

Lie-stable subspaces are stable. Since $k$ has characteristic $0$ and $G$ is connected and smooth, a subspace $W$ of a rational representation with $\mathfrak{g}W\subseteq W$ is $G$-stable (Lie algebras of subspace stabilizers and Lie-stable subspaces).

Semisimple groups have no characters. $X(G)=0$, so a one-dimensional rational representation of the semisimple group $G$ is trivial (Semisimple groups are perfect and have no nontrivial characters).