Let $X\sim\mathcal N(\mu,\Sigma)$ for some $\mu\in\mathbb R^d$ and a symmetric, positive semidefinite $\Sigma\in\mathbb R^{d,d}$, i.e. $\varphi_X(t)=\exp(it^T\mu-\frac12t^T\Sigma t)$ for every $t\in\mathbb R^d$.
Using the characteristic function I could easily show that for every $E\sim\mathcal N(0,I_d)$ and every $A\in\mathbb R^{d,d}$ satisfying $\Sigma=AA^T$ we have $X\sim AE+\mu$, but I could not prove that the components of $E$ are independent or that there is at least one $A\in\mathbb R^d$ and $E=(e_1,\dots,e_d)$ such that $e_1,\dots,e_d\sim\mathcal N(0,1)$ are independent and $X\sim AE+\mu$ as the second definition here states.
Can you guys give me some hints or can you suggest some literature that might help me?
Random variables with a joint normal distribution are independent iff their covariances are $0$. Since $E(E_iE_j)=0$ for $i\neq j$ and $E(E_i)=0$ for all $i$ it follows that the components $E_i$ are independent.