Let $u,v:\mathbb{R}^m\to\mathbb{R}^3$ be $C^1$ functions. I need to prove the following identity: $$ \langle \nabla u, \nabla v \rangle = \langle u,u \rangle \langle \nabla u, \nabla v \rangle - \langle u,\nabla v \rangle \cdot \langle u, \nabla u \rangle, $$
where $\langle\cdot, \cdot \rangle$ denotes inner product which is supposed to be understood like this:
$$\langle \nabla u, \nabla v \rangle := \sum_{j=1}^{m}\sum_{\alpha,\beta=1}^{3} \frac{\partial u^\alpha}{\partial x_j}\frac{\partial v^\beta}{\partial x_j},$$ $$\langle u, \nabla u \rangle := \left( \sum_{\alpha=1}^{3} u^\alpha \frac{\partial v^\alpha}{\partial x_j}\right)_{j=1,\ldots,m}$$
So $$ \langle u,\nabla v \rangle \cdot \langle u, \nabla u \rangle = \sum_{j=1}^{m} \sum_{\alpha,\beta=1}^{3} u^\alpha \frac{\partial v^\alpha}{\partial x_j}u^\beta\frac{\partial u^\alpha}{\partial x_j}$$
I got stuck with those indices several times now and I am starting to believe this identity may not hold at all...
Can anyone see how to prove this? I'd appreciate if there were an elegant way to do so (without playing with indices).
Thanks in advance, Kamil