How can I show that the cartesian product of the Lie algebras $\mathfrak{so}(5)$ (type $B_2$) with $\mathfrak{sl}(2)$ (type $A_1$) is isomorphic to a subalgebra of the Lie algebra of type $F_4$?
I'm using $[(x,y),(z,w)]=([x,z],[y,w])$.
I've been trying to construct the Dynkin diagram of $\mathfrak{so}(5)\times\mathfrak{sl}(2)$, but I feel there's an easier argument.
Well any subdiagram of the Dynkin diagram gives a root subsystem and thus a Lie subalgebra. This is fairly straightforward to prove with a little familiarity with root systems.
In fact we can take this further. You can extend the diagram by adding in a node corresponding to the lowest root (or the lowest short root) before you take a subdiagram. In $F_4$ these extended diagrams are those with an extra node at one of the ends. You can then (with either of these diagrams) find $B_2\times A_1$ as a subdiagram. The $A_1$ part will be the new node and the $B_2$ will be the two nodes in the middle of the original diagram.
Note in one case the $A_1$ part is given by a short root and in the other case by a long root so these won't be conjugate for example.