Suppose $X$ and $Y$ are finite, simply connected, based CW-complexes and $m,n\geq 2$. If $a\in \pi_m(X)$ and $b\in \pi_n(Y)$, then one can regard these as elements of the homotopy groups of $X\vee Y$. If $a$ and $b$ are non-trivial, must the corresponding Whitehead product $[a,b]\in \pi_{m+n-1}(X\vee Y)$ be non-trivial?

I understand some cases using homotopy excision and the smash product but not this general situation.