Euler angles of rotation about x=y=z

[Null-Set]

What would be the euler angles of rotation 2pi/3 about the line x=y=z? If something were in the xy plane and it underwent that rotation, would it end up in the yz plane?

 

[mnb96]

I am not totally sure about this but as a starting point you could consider the rotation matrix with Euler angles $$R_{\alpha,\beta,\gamma}$$, and a transformation matrix $$Q$$ which rotates around the axis x=y=z of $$2\pi / 3$$ radians. Then for any vector $$v \in \mathcal{R}^3$$, you essentially want to solve for $$\alpha,\beta,\gamma$$ the following system:

$$R_{\alpha,\beta,\gamma}v=Qv$$

Also, try to take a look here: http://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles because you might already find the solution you were looking for. I suspect you are essentially looking for a conversion from quaternion representation to Euler angles.
Hope it helped.