Is there any references on LU decomposition for orthogonal or unitary matrices?

It seems to me that the diagonal entries of $U$ has some nice structure regarding to the Euler angles of the original matrix. As one can easily see under a Euler parametrisation:

$$\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}=\begin{bmatrix}1&0\\\tan\theta&1\end{bmatrix}\begin{bmatrix}\cos\theta&\sin\theta\\0&1/\cos\theta\end{bmatrix}.$$ And for the $3\times 3$ case, the diagonal entries for $U$ should be something similar to $$\cos\theta_1\cos\theta_2, \cos\theta_3/\cos\theta_1, 1/\cos\theta_3\cos\theta_2.$$ Is there any previous work on these?