Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus of $G$ with roots $\Phi$, and $R(G)$ the radical of $G$. Then $G/R(G)$ and the derived group $[G,G]$ are semisimple groups. I'm interested in relating the roots of $G$ to those of $G/R(G)$ and $[G,G]$. My first question is:
1 . We know that $T/R(G)$ is a maximal torus of $G/R(G)$. Is $T \cap [G,G]$ a maximal torus of $[G,G]$? This is definitely the case when the Weyl group of $G$ has order $2$.
But my main question is:
2 . How can the roots in $G$ be identified with roots in $[G,G]$ and $G/R(G)$?
Here is what we can say about $G/R(G)$. If $S$ is a subtorus of $T$, there is a homomorphism of cocharacter groups $Y(T) \rightarrow Y(T/S)$ given by $\lambda \mapsto \pi \circ \lambda$, where $\pi$ is the canonical map $T \rightarrow S$. The kernel is those cocharacters of $T$ whose image is contained in $S$, i.e. $Y(S)$. Thus $Y(T)/Y(S)$ injects into $Y(T/S)$, and since these groups have the same rank, we have the following:
There exists an integer $m \geq 1$, such that for any $\gamma \in Y(T/S)$, there exists a $\lambda \in Y(T)$ such that $\lambda(x)S = n\gamma(x)$ for all $x \in k^{\ast}$.
Now take $S = R(G)$, which is the identity component of the center of $G$, equivalently the identity component of the intersection of kernels of roots of $T$ in $G$. Then $X(T/R(G))$ injects into $X(T)$, and all the roots $\alpha \in \Phi$ lie in $X(T/R(G))$. We see that $\Phi(G,T) = \Phi(G/R(G),T/R(G))$ under this identification $X(T/R(G)) \subseteq X(T)$.
And in $X(T/R(G))$, the span $E$ of the roots $\alpha \in \Phi$ has the same rank as that of $X(T/R(G))$: if not, then $E^{\perp} = \{ \gamma \in Y(T/R(G)) : \langle \alpha, \gamma \rangle = 0, \textrm{ for all } \alpha \in \Phi \}$ has dimension at least one. But if $\gamma \in E^{\perp}$, write $m \gamma = \pi \circ \lambda$ for some $\lambda \in Y(T)$. As $\alpha \circ \gamma$ is the identity on $k^{\ast}$ for all $\alpha \in \Phi$, the same is true of $\lambda$, whence the image of $\lambda$ is contained in the intersection of all the kernels $\textrm{Ker } \alpha$, hence the image of $\lambda$ is contained in $R(G)$. Thus $m \gamma$, and hence $\gamma$, is $0$.
This argument, combined with the difficult integrality theorem and a couple of other things, tells us that $X(T/R(G)) \otimes_{\mathbb{Z}} \mathbb{R}$ together with the roots $\Phi$ is a root system.
That covers the description of roots of a maximal torus in $G/R(G)$. Now, what can be said about the semisimple group $[G,G]$?