I asked this question in math.stackexchange two days ago, but no one has answered yet. I suspect it is "hard enough" that it is appropriate to post it here as well. I am new to stackexchage, so if it is not appropriate to post the same question twice please let me know.
It is known that any rotation in $SO(n)$ can be decomposed into a particular sequence of $n(n-1)/2$ simple rotations (that is, rotations which rotate a 2D plane in $\mathbb{E}^n$). The procedure to construct an arbitrary $SO(4)$ rotation in this manner is illustrated in the answer to: https://math.stackexchange.com/questions/365207/decomposition-of-a-single-4d-rotation. I do not know whether there is a name for these decompositions, so will refer to them as Givens decompositions. I am interested in whether a different kind of decomposition, which I describe below, is also possible for an arbitrary element of $SO(4)$.
To describe the Givens decomposition in $4D$, let us fix an orthogonal set of coordinates $x,y,z,w$. Then the Givens decomposition for some $T \in SO(4)$ has the form $T = T_{zw}T_{yw}T_{xw}T_{yz}T_{xz}T_{xy}$, where $T_{ij}$ is a rotation which only rotates the plane spanned by the coordinates $i$ and $j$, leaving the vectors orthogonal to this plane invariant. It is easy to see that this decomposition works by right multiplying an arbitrary $R \in SO(4)$ by $T$ and choosing $T_{zw}$ such that it would erase the $R_{43}$ element, then choosing $T_{yw}$ to erase the $R_{42}$ element, and so on. So each simple rotation in the Givens decomposition of $T$ erases one of the lower off-diagonal elements of $R$, and since the resulting matrix must be in $SO(4)$, you are left with the identity.
The question I am interested in is whether a decomposition of the form $T = T_{xz}T_{yw}T_{xw}T_{yz}T_{xy}T_{zw}$ exists. Notice that each consecutive pair in the sequence is composed of commuting rotations (since, e.g., $T_{xz}$ and $T_{yw}$ act on orthogonal planes). So the form is not unique even after fixing the coordinate axes.
I have tried following the same logic as used to prove the existence of a Givens decomposition, but the equations became a bit too complicated too fast. I was wondering if there is an easier way to see if such a decomposition always exists.
Remark: This is related to the question of characterization of the set of left cosets of the symplectic subgroup of $SO(4)$. If the aforementioned decomposition exists, then any element of $SO(4)$ can be written as $T_1T_2T_s$. Here $T_s$ is symplectic, while $T_1$ and $T_2$ are generated by matrices $X_1$ and $X_2$, respectively, which span the space orthogonal to the Lie algebra of the symplectic subgroup of $SO(4)$.
