About Goto's method showing surjectivity of word maps

Let F be the free group on two letters. For {\omega} \in F we study the associated word map {\omega}: SU(n) \times SU(n) \rightarrow SU(n). Extending a method of Goto, we show that for {\omega} not in the second derived subgroup F^(2) of F, there are infinitely many n \in N such that the associated word map {\omega} : SU(n) \times SU(n) \rightarrow SU(n) is surjective.