I asked this as subquestion in a comment pursuant to my Banach-Tarski question. I think it is worth promoting here to a question in its own right.

Consider these two matrices over ${\Bbb R}[[\epsilon]]$: $$A = \left[ \begin{array}{ccc} \cos(\epsilon) & \sin(\epsilon) & 0 \\ -\sin(\epsilon) & \cos(\epsilon) & 0 \\ 0 & 0 & 1 \end{array} \right] \ {\rm and}\ B = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos(\epsilon) & \sin(\epsilon) \\ 0 & -\sin(\epsilon) & \cos(\epsilon) \end{array} \right].$$

(By $\sin$ and $\cos$ I mean the formal Taylor series.)

If a word $w$ in $A$ and $B$ equals the identity modulo $\epsilon^k$, must $w$ belong to the $k$th term of the derived series of the free group on the symbols $A$ and $B$?