$\sigma=(1,2,4)(3,5)(6,7,8,9) \in S_9$. Find $\sigma_1, \sigma_2 \in S_9$ such that $\sigma=\sigma_1 \sigma_2 \neq \sigma_2 \sigma_1$ and $|\sigma_1|=3$ and $|\sigma_2|=4$.
Is there any trick to somehow manipulate disjoint cycle decompositions into joint ones? I find through brute force (and luck) that I can do this $\sigma=(124)(35)(6789)=(123)(324)(35)(6789)=(123)\cdot((5243)(6789))$. But how such problem may generally be solved.