@NarutakaOZAWA Thanks! This is a good point.

@NarutakaOZAWA This looks interesting, but it might be helpful if your notations ($e_k$, $f_i(k)$, overline, etc.) could be explained a little more.

@Malkoun That's a good point. In this case, A can be scaled to a projector. Whenever A is a projector, we have $A^p=A$ for any p, and this inequality always holds.

This is very awesome! It's quite surprising to me that the Lie subsuperalgebra generated by transpositions contains almost everything except for the centers.

@მამუკაჯიბლაძე Thanks! The function CanonicalBasis() is helpful for me. It seems for this particular example it is easy to get the semisimple type over $\mathbb{R}$ by combining the results from over $\mathbb{Q}$ and over $CF(24)$. I guess GAP would not be able to deal with matrices with transcendental elements, although I am not sure whether I will encounter such case.

@მამუკაჯიბლაძე May I know how did you get this in GAP? Thanks!

This is very elegant. Thank you so much!

@YCor I checked from 3 to 9.

@abx Thanks for the suggestion! I've made the change in the text.