Question
Define $\mathbf{a} = (a_1, \ldots, a_p)$ where $p$ is a positive integer and the $a_l$ are i.i.d $\text{Uniform}(-1,1)$ random variables. Fix a unit vector $w \in \mathbb{R}^p$. Consider the quantity $S = \mathbf{a}^T w = \sum_{j=1}^p a_j w_j$. Prove the following bound on the cumulant generating function: $$\log \mathbb{E}[\exp(3 \lambda S^2)] \leq 2\lambda^2 + \lambda, \quad \forall \lambda \leq 1/4.$$
Attempt
I am not sure how to handle the expectation of the squared sum, since $$S^2 = \sum_{j=1}^p \sum_{i=1}^p a_j a_i w_j w_i.$$
I was able to find the cumulant generating function of $S$ itself, and this is given by $$\log \mathbb{E}(\exp \lambda S) = \sum_{i=1}^p \left\{\log(e^{\lambda w_i} - e^{-\lambda w_i})- \log[2 \lambda w_i]\right\},$$
as can be seen from example from direct computation or adapting the result from this SE post while using the fact that the $a_j$ are iid so that the cumulative property of cumulant generating functions (see Section on 'Some Basic Properties' from Wikipedia) holds. I am not sure if that helps with the cumulants of $S^2$, which I am stuck on. How can one approach this problem?