Why is the set of commutators not a subgroup? I was surprised to see that one talks about the subgroup generated by the commutators, because I thought the commutators would form a subgroup. Some research told me that it's because commutators are not necessarily closed under product (books by Rotman and Mac Lane popped up in a google search telling me). However, I couldn't find an actual example of this. What is one? The books on google books made it seem like an actual example is hard to explain. 
Wikipedia did mention that the product $[a,b][c,d]$ on the free group on $a,b,c,d$ is an example. But why? I know this product is $aba^{-1}b^{-1}cdc^{-1}d^{-1}$, but why is that not a commutator in this group?

See the exercise 2.43 in the book "An introduction to the theory of group"- Joseph Rotman (4th ed.). 
He also had made a nice remark:
The first finite group in which product of two commutators is not a commutator has order 96.
