What is wrong in my attempt to rotate two orthogonal vectors? Let $V=(1,0,0)$ and $W=(0,1,0)$ be two vector. Consider the following transformation:
$\pi/4$ rotation around the $z$-axis $V$ and $W$ concurrent counterclockwise, then $\pi/4$ rotating the result concurrent in direction from $y$-axis to $z$-axis (around $x$-axis) with fixed origin.

I think the result vectors is $V^\prime=(1,1,1)$ and $W^\prime=(-1,1,1)$ up to positive rescaling. and the above transformation is angle-preserving. but $\left<V^\prime,W^\prime\right>\neq0$!!

What is wrong in my attempt?

By putting your pieces of clues together and some guesswork I reach the conclusion that the second transformation is no rotation at all. It looks like it's just about adding $e_z$ to the vector (which is a shearing transformation). 
The rotation preserves orthogonality all right even if you seem to be scaling the result as well. But the second does not, if $v$, $w$ and $e_z$ are orthogonal:
$$\langle v+e_z, w+e_z\rangle = \langle v,w\rangle + \langle v,e_z\rangle + \langle e_z, w\rangle + \langle e_z, e_z \rangle = \langle e_z, e_z\rangle = 1$$
This explains exactly how you get your result $\langle V', W'\rangle = 1\ne 0$
