  Let $\mathfrak{g}$ be a Lie algebra all of whose regular subalgebras of rank
2 are type $A_{1}\times A_{1}$, $A_{2}$, or $C_{2}$, and let $B$ be a crystal
graph corresponding to a representation of $\mathfrak{g}$. We explicitly
describe the local structure of $B$, confirming a conjecture of Stembridge.
