Show that matrices commute: if Exponential of linear combination of matrices is product of exponentials of the matrices

Question

In this question it is discussed how the exponential of a linear combination of two matrices reduces to the product of the exponentials of the two matrices, given that they commute. $$exp(aX+bY)=exp(aX)exp(bY)$$ However, this shows only one direction. How can I show that the matrices have to commute if the exponentials reduce in that way? It is easy to show that up to second order: I just expand the exponential on both sides can bring everything to one side. But it gets really complicated for higher orders, one gets nested commutators.