I'm currently trying to understand how we can derive the weights of a subalgebra of a given representation of a Lie group.
For example, if we start with the 16-dimensional representation of $SO(10)$, which corresponds to highest weight $(1,0,0,0,0,0)$, we have a long list with weights that define this representation. If we then consider, for example, the subalgebra $SU(5) \otimes U(1)$. How can I find the weights and therefore the representations for this subalgebra?