I am struggling with understanding the Jacobi identity (I am using it for matrix's commutator).
Let $[A,B] = AB - BA$. Then the following is true
$[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0$
Also the following is true
$[A,B] + [B,A] = 0$
But increasing the number of matrices I never got any identity..
$[A,[B,[C,D]]] + [B,[C,[D,A]]] + [C,[D,[A,B]]] + [D,[A,[B,C]]] \ne 0$
$[A,[B,[C,[D,E]]]] + [B,[C,[D,[E,A]]]] + [C,[D,[E,[A,B]]]] + [D,[E,[A,[B,C]]]] + [E,[D,[A,[B,C]]]] \ne 0$
Why is it true only when three matrices are involved?