Q:

Composition of plane rotations around different axes

Source:
Devide, Vladimir. Mathematical Problems and Expositions: 1) Tasks from Abstract Algebra
Prove that compositions of plane rotations around different axes don't form a group. (there is one plane rotated around different axes)
We haven't reached this point in our lectures and I'm rather intrigued by this. I know what properties a group has, but I don't know how a rotation composition of this kind behaves. 
So far, I' ve seen how the composition of rotations around the same axis can be represented by matrices and how the association can be interpreted as making either one step at a time or more steps at the same time.
Can anybody give me, if possible, a short insight into this specific problem?

A:

Experiment. Suppose you rotate the $xy$-plane about the $x$-axis and then rotate the resulting plane about  the $z$-axis. Can you get to that configuration by a single plane rotation from the initial configuration?

