In the Lectures on Riemann Surfaces by Otto Forster, he says to "Use the fact that every matrix $A \in \text{SO}(3)$ may be written as a product $A = A_1 \dots A_k$ where $$A_j = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1\\ 1 & 0 & 0 \end{bmatrix}$$ or else is a matrix of the form $$\begin{bmatrix} B & 0\\ 0 & 1 \\ \end{bmatrix}$$ where $B \in \text{SO}(3)$."
I have never worked with this group before so I'm unfamiliar with this kind of decomposition. I'm also unsure about what he means by the quoted section. Are all orthogonal matrices either of the "B- form" or such a product or are all orthogonal matrices a product of several of each? How would one go about proving such a fact? It seems related to Gauss-Jordan decomposition but I've been having trouble proving it. Thank you!
The phrasing is confusing, as one would tend to interpret "or else is" as an alternative to "may be written" (with subject $A$) and not to the equals sign in the preceding equation (with subject $A_j$); but that's how it's meant. So each $A_j$ is either the given permutation matrix or a rotation about the $z$ axis. Using the permutation matrix $P$ and a rotation $R$ about the $z$ axis, you can form rotations about the other two axes: $PRPP$ is a rotation about the $y$ axis, and $PPRP$ is a rotation about the $x$ axis. Then the composition follows from the fact that all rotations can be written as a product of rotations about the coordinate axes (see e.g. Euler angles).