$S_{11}+S_{12}$ can be negative, so it is never a multiple of a chi-square

Let $U_1,U_2,\dots,U_n$ be independent identically distributed random vectors with common distribution $N_p(0,\Sigma)$, $\Sigma=((\sigma_{ij}))$ is a positive definite matrix. Let $S=((S_{ij}))=\sum_{j=1}^nU_jU_j'$. Then which of the following statements is not true?