For anyone confused, he says that a 4-element set can be split into two pairs in exactly 3 ways, e.g. { { 1, 2 }, { 3, 4} }, { { 1, 3 }, { 2, 4 } }, { { 1, 4 }, { 2, 3 } }. Any permutation of the 4-element set naturally induces a permutation on these 3 partitions: if f : { 1, 2, 3, 4 } -> { 1, 2, 3, 4 } is a permutation, then the induced permutation maps { { a, b }, { c, d } } to { { f(a), f(b) }, { f(c), f(d) } }. This assignment is a non-trivial homomorphism from permutations on 4 elements to permutations on 3 elements, and the kernel of that homomorphism is a witness that S_4 is not a simple group. Somehow this further proves that SO(4) is not a simple Lie algebra, but I didn't follow that part.
This is only just begining: I can’t immediately tell if this is C*C = (C) * (C) or R^4 = (R^3) * (R) Neither of which has to do with what this is probably talking about, which is the technically 4 dimensional TM = 2 + 2 in a way that’s over the real as it were but is sincerely different than what can be topologically deduced is the natural implication of the previous demonstrations of explicit geometry. An exact comparison must be given to this kind of thing
