Consider the following SDE: $$X(t)= s + \int_{t_0}^t b(X(s)) dt + \int_{t_0}^t a(X(s)) dW(t) $$
where $X(t_0)=s$ and $a:\mathbb{R}^d \rightarrow \mathbb{R}^{d \times N} $,$b:\mathbb{R}^d \rightarrow \mathbb{R}^d $ and $W$ denoting an $N$ dimensional Brownian Motion.
The infinitesimal generator is defined then as $$L[u](x) = b(x) D_x u(x) + \frac{1}{2} tr[a(x)a(x)^T D^2_x u(x)]$$
I do not understand why in this formula the mixed second derivatives will not be considered, meaning something like $\partial_{x_i x_j}$ for $i \ne j $
An application of Itos-formula on $du(X(t)))$ yields $$du(X(t))) = L[u](X(t))dt + D_x u(X(t)) a(x) dW(t)$$
Why do I not need the mixed second derivatives in this formula matching with the defintion of the infinitesimal generator?