I'm having the same problem as it was questioned [here.](https://math.stackexchange.com/questions/109917/existence-of-a-local-geodesic-frame)

I can't get throught the step where I need to show that $\nabla\_{E\_i}E\_j (p)=0$. It only leads to
$$
\nabla\_{E\_i}E\_j(p)=\sum\_{lk}^n a\_{il}(0)\dfrac{\partial b\_jk}{\partial x\_l}(0)\partial x\_k(p)
$$
which I can't show that equals zero.

On the linked question, there is also the indication that $\nabla\_{E\_i}E\_j (E\_k)=\Gamma^k\_{ij}$. This could lead that $\nabla\_{E\_i}E\_j (E\_k)(p)=\Gamma^k\_{ij}(p)=0$, but $\nabla\_{E\_i}E\_j (E\_k)(p)$ is not equal to $\nabla\_{E\_i}E\_j (p)$ which is the one I need.

Can someone help me?