Canonical forms for a class of pairs of commuting nilpotent matrices under simultaneous similarity

We present canonical forms for all indecomposable pairs $(A,B)$ of commuting nilpotent matrices over an arbitrary field under simultaneous similarity, where $A$ is the direct sum of two Jordan blocks with distinct sizes. We also provide the transformation matrix $X$ such that $(A,X^{-1}BX)$ is in its canonical form.