Proofs for certain ways of decomposing permutations as products of transpositions I know $(1 2 3 4 5) = (15)(14)(13)(12)$. But I just discovered $(12345) = (12)(23)(34)(45)$ and $(12345) = (54)(52)(21)(25)(23)(13)$. Also, $(15) = (21)(32)(43)(54)(43)(32)(21)$. Excepting the first one that I know already, where do these methods of permutation decompositions come from? I couldn't find their proofs.

To compose permutations, you can track what happens to the individual elements. For example, to check $(12345)=(54)(52)(21)(25)(23)(13)$, apply each transposition, starting on the right:

$$ 12345\\\ 32145\\\ 23145\\\ 53142\\\ 53241\\\ 23541\\\ 23451 $$

The result is the same as that of applying $(12345)$.