Calculating the determinant of a 4×4 matrix using 2×2 minors. What is the +/- pattern in this case?

I remember that in the past a few times I tried to write determinants of say a nxn matrix in determinants of blocks of that matrix. It always failed but now I understand the way the matrix version of the theorem of Pythagoras goes, all of a sudden it is a piece of cake.
In this post I only give an example of a 4×4 matrix. If you take the first two columns of such a matrix, say AB, this is now a 4×2 matrix with six 2×2 minors in it.
If we name the last two columns as CD, for every 2×2 minor in AB there is a corresponding complementary 2×2 minor in CD. For example if we pick the left upper 2×2 minor in our matrix ABCD, it’s complement is the right lower 2×2 minor at the bottom of CD.
If you take the determinants of those minors and multiply them against the determinants of their complements, add it all up with a suitable +/- pattern and voila: that must be the determinant of the whole 4×4 matrix.

This method could more or less easily expanded to larger matrices, but I think it is hard to prove the +/- pattern you need for the minors of larger matrices. Because I am such a dumb person I expected that half of my six 2×2 minors pick up a minus sign and the other half a plus sign. Just like you have when you develop a 4×4 determinant along a row or column of that matix. I was wrong, it is a bit more subtle once more confirming I am a very very dumb person.

I skipped giving you the alternative way of calculating determinants: The determinant is also the sum of so called signed permutations on the indices of their entries. If you have never seen that I advice you to look it up on the internet.

Because I skipped excisting knowledge widely available already, I was able to do the calculation in just four images! So it is a short post. Ok ok I also left out how I did it in detail because writing out a 4×4 determinant already has 24 terms with each four factors. That’s why it is only four pictures long in this post…

(I later replaced the above picture because it had a serious typo in it.)

If you want you can also use that expression as the determinant as a sum of signed permutations. It is a very cute formula. Wiki title:
Leibniz formula for determinants.
And a four minute video, it starts a bit slow but the guy manages to put in most of the important details in just four minutes:

Ok, that was it for this post. I filed it under the category ‘Pythagoras stuff’ because I tried similar stuff in the past but only with the knowledge into the matrix version of the Pythagoras theorem, it all becomes a bit more easy to do.

Thanks for your attention.