  In this article, we classify all standard invariants that can arise from a
composed inclusion of an $A_3$ with an $A_4$ subfactor. More precisely, if
$\mathcal{N}\subset \mathcal{P}$ is the $A_3$ subfactor and
$\mathcal{P}\subset\mathcal{M}$ is the $A_4$ subfactor, then only four standard
invariants can arise from the composed inclusion
$\mathcal{N}\subset\mathcal{M}$. This answers a question posed by Bisch and
Haagerup in 1994. The techniques of this paper also show that there are exactly
four standard invariants for the composed inclusion of two $A_4$ subfactors.
