When $2$-cycles (transpositions) are not disjoint, they don't necessarily commute?

Question

Consider a permutation that can be written as a product of $2$-cycles, for instance: $$ (1632)(457)=(12)(13)(16)(47)(45) $$

The $2$-cycles on the right-hand side of the identity above don't necessarily commute, correct? Since they are not disjoint.

Answer 1

$(a\,b)(b\,c)=(a\,b\,c),\qquad (b\,c)(a\,b)=(a\,c\,b)=(a\,b\,c)^{-1}$.