Show that the product of two transpositions can be expressed as a product of $3$-cycles

Question

Consider the symmetric group $S_n$ where $n>2.$

Show that the product of two transpositions $(ab),\,(cd)$ can be written as a product of $3$-cycles where $a,b,c,d$ are all distinct.

I'm not sure where a sensible place to start is so any suggestions would be appreciated.

Answer 1

Try $(ab)(cd) = (ab)(bc)(bc)(cd)= (abc)(bcd)$.

Answer 2

$(abc)$ maps $a$ to $b$, $b$ to $c$, and $c$ to $a$. Now, all we need to do is map $c$ to $a$ (so $b \mapsto c \mapsto a$), $a$ to $d$ ($c \mapsto a \mapsto d$), and $d$ to $c$ — i.e., $(cad)$. Thus, $(ab)(cd) = (cad)(abc)$.