In 4 dimensions, this is not true. A rotation in XY cannot compose with one in ZW so at least two rotations are required.
Here's an example that's easiest to sort-of-visualize:
1. Rotate the y-z plane of 4-d space by half a turn (180 degrees). That is, transform every point in 4-space according to w->w, x->x, y->-y, z->-z.
2. Rotate the w-x plane, also a half-turn. So w->-w, x->-x, y->y, z->z.
Composing these two transformations yields w->-w, x->-x, y->-y, z->-z. That composed transformation is an inversion, not a rotation. Is it clearer now? (It was late when I wrote last night, sorry. Let me try and rephrase how I defined a rotation: a linear transformation that carries one unit vector to a second unit vector, and leaves unchanged any vectors that are orthogonal to both. In my previous comment the plane I was talking about was the plane that contains both given vectors. Of course, the null rotation doesn't pick out any particular plane, though it's not otherwise special in these terms. IANAM, but would appreciate correction if I'm goofing.)