The solution to this exercise is nicely depicted in Figure 41.4 of reference [1], which is attached.

Attachment:

Misner,Thorne,Wheeler Fig.41.4.GIF [ 6.47 KiB | Viewed 1484 times ]
In this picture, a rotation is first performed around the rotation axis pointing in z-direction with an angle Theta_1, followed by a rotation around the axis pointing in approximate y-direction with an angle Theta_2.

In analogy to Fig.11.4 of Penrose's book, a triangle can be constructed on the unit sphere showing the relation of the two consecutive rotations and the resulting composite rotation. The vertices of this triangle will be those where the three rotation axes intersect the unit sphere.

One side of the triangle is given by the two vertices corresponding to the first and second rotations. The other two sides, and thus vertex "3" are defined by the angles (-1/2 Theta_1) and (+1/2 Theta_2) at vertex "1" and "2", respectively.

The third vertex, "3", then corresponds to the rotation axis of the composite rotation, and the rotation angle of the composite rotation is two times the angle between the grand arcs given by vertices "1"-"3" and "2"-"3" (1/2 Theta_net in the picture).

That all this is correct, can easily be seen by using the insight gained from Exercise [11.6]. From there we know, that any 3D-rotation can be decomposed into two consecutive reflections on two planes, where the intersection of those planes is the rotation axis and the angle between the planes is half the rotation angle. Note, that the two planes can be freely rotated about this axis, it is only the angle between them that matters.

The sides "1"-"3" and "1"-"2" of the triangle on the unit sphere define just these two mirror planes for the first rotation and the sides "1"-"2" and "2"-3" for the second rotation. The possibility to rotate these planes freely around the rotation axes is used here to make the second mirror plane of the first rotation and the first mirror plane of the second rotation coincide. When composing the two rotations, or four consecutive reflections, the second and the third reflection, both "1"-"2" cancel each other, and only the reflections "1"-"3" and "2"-"3" remain.

[1] C. W.Misner, K.S.Thorne, J.A.Wheeler, Gravitation (Freeman 1973)