I am going through a book (Roberts and Schmidt, *Local Newforms for GSp(4)*), which states the following group decomposition. Let $F$ be a non-archimedean field, $\mathfrak{o}$ its ring of integers and $\mathfrak{p}$ its maximal ideal. Then, in terms of subgroups of $\mathrm{GSp}(4)$,
$$\left(
\begin{array}{cccc}
\mathfrak{o}&\mathfrak{o}&\mathfrak{o}&\mathfrak{o}\\
\mathfrak{p}^n&\mathfrak{o}&\mathfrak{o}&\mathfrak{o}\\
\mathfrak{p}^n&\mathfrak{o}&\mathfrak{o}&\mathfrak{o}\\
\mathfrak{p}^n&\mathfrak{p}^n&\mathfrak{p}^n&\mathfrak{o}
\end{array}
\right)
=
\left(
\begin{array}{cccc}
1& & & \\
\mathfrak{p}^n&1& & \\
\mathfrak{p}^n& &1& \\
\mathfrak{p}^n&\mathfrak{p}^n&\mathfrak{p}^n&1
\end{array}
\right)
\left(
\begin{array}{cccc}
\mathfrak{o}^\times&&&\\
&\mathfrak{o}&\mathfrak{o}&\\
&\mathfrak{o}&\mathfrak{o}& \\
& & &\mathfrak{o}^\times
\end{array}
\right)
\left(
\begin{array}{cccc}
\mathfrak{1}&\mathfrak{o}&\mathfrak{o}&\mathfrak{o}\\
&\mathfrak{1}& &\mathfrak{o}\\
& &\mathfrak{1}&\mathfrak{o}\\
& & &\mathfrak{1}
\end{array}
\right)
$$

where missing entries are zeros. I would like to understand this decomposition more generally, for I would like to apply that for other groups. Is there any general setting for this Iwahori factorization? (it seems similar to an $LU$ factorization, however is it always true and what are exactly the three groups appearing on the right?)

I got only partial answers on MSE hence I post the question here.