I have $A$, $B$, $C$, and $D$ i.i.d normal random variables of mean 0 and variance 1 and the following transformations where $U$, $V$, $W$, $\theta$ and $G_{11}$, $G_{12}$, $G_{21}$, and $G_{22}$ are random variables $$ \left[\begin{array}{ll} \boldsymbol{A} & \boldsymbol{B} \\ \boldsymbol{C} & \boldsymbol{D} \end{array}\right]=\left[\begin{array}{cc} \cos \boldsymbol{\theta} & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{ll} U & V \\ 0 & W \end{array}\right] $$
and $$ \left[\begin{array}{ll} \boldsymbol{G}_{11} & \boldsymbol{G}_{12} \\ \boldsymbol{G}_{12} & \boldsymbol{G}_{22} \end{array}\right]=\left[\begin{array}{ll} \boldsymbol{A} & \boldsymbol{B} \\ \boldsymbol{C} & \boldsymbol{D} \end{array}\right]^{T}\left[\begin{array}{ll} \boldsymbol{A} & \boldsymbol{B} \\ \boldsymbol{C} & \boldsymbol{D} \end{array}\right]=\left[\begin{array}{ll} \boldsymbol{U} & \boldsymbol{V} \\ \boldsymbol{0} & \boldsymbol{W} \end{array}\right]^{T}\left[\begin{array}{ll} \boldsymbol{U} & \boldsymbol{V} \\ \boldsymbol{0} & \boldsymbol{W} \end{array}\right] $$ I would like to find the PDF of $U$, $V$, and the conditional PDF of $G_{12}$ given $G_{11}$ and $G_{22}$.
The problem is that I don't even know how to proceed. One way is to go with Jacobian but the issue is how to inverse the matrices of RV? Any suggestion??