Let $w=w(a,b)$ be a non-trivial word in the free group $F_2 = \langle a,b \rangle$ and $w_G \colon G \times G \to G$ be the induced word map for some compact Lie group $G$.
Murray Gerstenhaber and Oskar Rothaus showed in 1962 (see M. Gerstenhaber and O.S. Rothaus, The solution of sets of equations in groups, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1531–1533.) that $w_G$ is surjective in a very strong sense for $G=U(n)$ whenever the exponent-sum of $a$ or $b$ in $w$ is non-zero. Using degree-theory and the calculations of the homology of compact Lie groups due to Heinz Hopf, they showed that if the exponent sum of $a$ is non-zero, then $u \mapsto w_{U(n)}(u,v_0)$ is surjective for any fixed $v_0 \in U(n)$.
Earlier, in 1949, Morikuni Gotô (J. Math. Soc. Japan, 1949 vol. 1 pp. 270-272) already showed that the commutator $w=[a,b]$ induces a surjective map $w \colon G \times G \to G$ for any simple compact Lie group $G$. His proof proceeded as follows: Since the commutators are a conjugation invariant set, one can assume that $u$ is in a fixed maximal torus. We can now take a suitable $a$ in that torus and $b$ an element the Weyl group $W_G$ of this torus, such that $b$ does not have $+1$ as an eigenvalue (when acting on the universal cover of the maximal torus). This implies that the action of $1-b$ on the universal covering is an isomorphism. Hence, every element in the maximal torus can be obtained as $a \cdot ba^{-1}b^{-1}$ for some $a$. This strategy can be used for a few other words but does not lead any further.
Question: Are there any other techniques to show that $w_{G}$ is surjective for some $w$ and $G$?
The easiest word for which I cannot answer this is $w= [[a,b],[a^2,b^2]]$.
Question: Is the word-map $w_{PSU(n)}\colon PSU(n) \times PSU(n) \to PSU(n)$ surjective for the word $$w = [[a,b],[a^2,b^2]]?$$
Best Answer
In this paper of Borel it is shown that any non-trivial word is a dominant map from $G \times G$ to $G$ whenever $G$ is a compact connected semisimple Lie group.
So the answer to your second question is "yes".I don't recall offhand exactly what techniques are used, but I vaguely recall that one starts with the $SL_2$ case (which contains free subgroups that one can play with) and builds up from there.EDIT: Dominance would imply surjectivity (or at least that the image is Zariski dense) in an algebraically closed field, but I didn't realise that the question is over the reals, and so Borel's result does not fully resolve the question.