Let $G$ be a connected algebraic group over an algebraically closed field $k$. If $G$ is reductive, then there is a very nice structure of the Lie algebra of $G$ as a direct sum of the Lie algebra of a maximal torus and of one dimensional "root spaces," which come in pairs. I was wondering if anything could be said about a converse.
Specifically, suppose $T$ is a maximal torus of $G$, and $Z_G(T) = T$, and assume the nonzero weights of $T$ under the rational representation $\textrm{Ad}: T \rightarrow \textrm{GL}(\mathfrak g)$ come in pairs $\alpha, - \alpha$, and each weight space $\mathfrak g_{\alpha}$ is one dimensional. Can we conclude from here that $G$ is reductive?
The example I'm keeping in mind is $\textrm{Sp}_4$. I have verified all these properties for $\textrm{Sp}_4$, but I have never actually gone to the trouble of showing that $\textrm{Sp}_4$ is reductive. I was wondering whether what I have done so far is actually enough. Or, are there a couple of other innocuous things to check in order to conclude $R_u(G) = 1$?