My question is related to this post. I am trying to compare the Weyl groups associated to root systems of type $D_2$ and $A_3$ respectively.
I know that the simple roots of $D_2$ are $e_1-e_2$ and $e_1+e_2$, and that the simple roots of $A_3$ are $e_1-e_2$, $e_2-e_3$ and $e_3-e_4$. The Weyl group $W'$ of the former root system can be written as $\{1,-1,s,-s\}$ where $-1$ is the sign change and $s$ is the permutation. The Weyl group of the latter is just $W\cong S_4$.
Is there a canonical way of embedding $W'$ in $W$? My question comes from the fact that I want to study $\text{SO}_4\subset \text{GL}_4$, where $\text{SO}_4$ is the orthogonal group where each matrix has determinant $1$. As there is an inclusion of reductive groups, there should be an inclusion of Weyl groups.
Many thanks in advance.