TITLE: A curious property of orthogonal matrices
QUESTION [17 upvotes]: Let $A$ be an $n\times n$ orthogonal matrix, i.e. $AA^T=A^TA=I$. I noticed experimentally that if we take any increasing set of indices $i_1<\cdots<i_p$, then the sum of the squares of all possible minors with rows corresponding to $\{i_1,\ldots,i_p\}$ is equal to 1.
This is a bit hard to formulate, so let me try to say what I mean more explicitly. Let
$$A=\begin{pmatrix}a_1^1&\ldots&a_1^n\\\vdots&\ddots&\vdots\\a_n^1&\cdots&a_n^n\end{pmatrix}$$
and let
$$A_{i_1\ldots i_p}^{j_1\ldots j_p}=\begin{pmatrix}a_{i_1}^{j_1}&\ldots&a_{i_1}^{j_p}\\\vdots&\ddots&\vdots\\a_{i_p}^{j_1}&\cdots&a_{i_p}^{j_p}\end{pmatrix}$$
for $i_1<\cdots<i_p$, $j_1<\cdots<j_p$ and $1\leq p\leq n$. 

Conjecture: If $A$ is orthogonal, then, for any $i_1<\cdots<i_p$ we have
  $$\sum_{j_1<\cdots<j_p}\left(\det A_{i_1\ldots i_p}^{j_1\ldots j_p}\right)^2=1,$$
  where the sum is over all increasing sets of $p$ indices.

How do we prove such a thing? Is that even true? (The answer is very likely yes, as I have checked it for many random examples.)

Note: The case $p=1$ is the statement that the rows of $A$ have unit norms, and the case $p=n$ is the statement that $(\det A)^2=1$. The conjecture generalizes these two statements.

In particular, the conjecture is clearly true for $n=2$. Now, here is a slightly less trivial example:
Example: Take
$$
A=
\begin{pmatrix}
1/\sqrt{2}&0&1/\sqrt{2}\\
-1/\sqrt{6}&\sqrt{2/3}&1/\sqrt{6}\\
1/\sqrt{3}&1/\sqrt{3}&-1/\sqrt{3}
\end{pmatrix}\in O(3).
$$
Then, with $p=2$ and $i_1=1$, $i_2=2$ we have
$$
\begin{vmatrix}
1/\sqrt{2}&0\\
-1/\sqrt{6}&\sqrt{2/3}
\end{vmatrix}^2
+
\begin{vmatrix}
1/\sqrt{2}&1/\sqrt{2}\\
-1/\sqrt{6}&1/\sqrt{6}
\end{vmatrix}^2
+
\begin{vmatrix}
0&1/\sqrt{2}\\
\sqrt{2/3}&1/\sqrt{6}
\end{vmatrix}^2
=\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1.
$$

REPLY [5 votes]: Following the Darij's comment. Let $B$ be the $p\times n$ submatrix of $A$ consisting of the lines $i_1,\cdots,i_p$ of $A$. Then $BB^T=I_p$ and $\det(BB^T)=1$. Thus $V=\sqrt{1}=1$ is the $p$-dimensional volume of the parallelotope $\Pi$ spanned in $\mathbb{R}^n$ by the $p$ rows of $B$. According to Cauchy Binet, $V^2$ is also  the sum of the squares of the volumes of the orthogonal projections of $\Pi$ onto the $\binom{n}{p}$ $p$-dimensional coordinate planes.