I am reading Do Carmo, and in the context of the Levi-Civita connection, he claims that the formula $$\sum_{\ell} \Gamma_{ij}^{\ell} g_{\ell k} = \frac 12 \left ( \frac{\partial}{\partial x_i} g_{jk} + \frac{\partial}{\partial x_j} g_{ki} - \frac{\partial}{\partial x_k} g_{ij} \right ), \tag{1}$$ where $\Gamma_{ij}^{\ell}$ are the Christoffel symbols and $g_{ij} = \langle \frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j} \rangle,$ follows easily from the the formula $$\langle \nabla_X Y, Z \rangle = \frac 12 ( X \langle Y, Z \rangle + Y \langle Z, X \rangle - Z \langle X, Y \rangle - \langle [X,Z], Y \rangle - \langle [Y,Z], X \rangle - \langle [X,Y], Z \rangle ). \tag{2}$$ Now, I think I must have some fundamental misunderstanding, because formula $(1)$ doesn't even make sense to me formally. The $g_{ij}$ are functions from the manifold to $\mathbb{R}$, and the Christoffel symbols are functions from the manifold to $\mathbb{R}$, so the LHS is a function from the manifold to $\mathbb{R}$. The RHS is a linear combination of tangent vectors, so it is a tangent vector. So I don't see how the LHS and RHS are even the same type of object.
Could you please clarify my confusion about formula (1)?