To find a rotation that would take the axes x → y, y → z and z → x two generators are required and the calculation proceeds as follows.
To find out what happens if we let the change occur incrementally can allow the two angles to change from 0° to 360 ° at a steady rate and compute a set of X, Y and Z values that indicate the position of the tips of the axes unit vectors.
We can view the results as an animated video. The red, green and blue colors indicate the x-, y- and z-axes respectively. Initially the rotation is difficult to follow but if one focuses on the blue z-axis one sees that it moves in a circle in the xz-plane while in the rotating x'y'-plane the x-axis always moves towards the y-axis.
Supplemental (Oct 6): I was able to upload a higher resolution video clip to YouTube and then add it to this post. Here's the link to the YouTube video. It is best viewed there in Theater mode. The color code is (R, G, B) represents (x, y, z). The same double rotation is generated if the second rotation is replaced by z → x.