Is it true that once a term in the Baker-Campbell-Hausdorff expansion is zero, then all the following terms must also be zero?
For example, it is not immediately obvious to me that if $[Y, [X, [X, Y]]]=0 $, then a term such as $[Y,[Y,[Y,[Y,X]]]]$ is zero too. Is there an easy way to see it?
It isn't true. For a semi-random example, take $$ X = \pmatrix{17 & 3\cr 6 & 9\cr},\ Y = \pmatrix{0 & 1\cr 2 & 3\cr} $$ Then $[Y,[X,[X,Y]]] = 0$ but $$[Y,[Y,[Y,[Y,X]]]] = \pmatrix{1156 & 867\cr 1734 & -1156}$$