 Solution

Expand the three increments into their eight monomial products. Exactly one term has every driving-sign index appearing an even number of times: take $a_{t-2}{a}_{t-1}$ from $X_t$, $a_{t-1}$ from $X_{t-1}$, and $a_{t-2}$ from $X_{t-2}$. Their product is $a_{t-1}^2{a}_{t-2}^2=1$. Every other term leaves an unpaired mean-zero sign. ThusIf the increments were independent, their three zero means would instead give zero. The nonzero third-order moment is therefore a direct demonstration of nonlinear temporal dependence despite vanishing nonzero-lag autocovariances.