Are Christoffel symbols structure coefficients? For a chart $(U,x^1,...,x^n)$ in an $n$ dimensional manifold, the the christoffel symbols for a covariant derivative are defined by $\nabla_{\partial_j}\partial_i=\Gamma_{ij}^k\partial_k$. For a general algebra of dimension $n$ with multiplication $\beta$, the structure constants are define by $\beta(x_i,x_j)=c_{ij}^kx_k$. Since the set of smooth vector fields is a vector space of $\mathbb{R}$, and the covariant derivative is bilinear, is this saying that the Christoffel symbols are exactly the structure constants of the algebra of smooth vector fields with the covariant derivative defined as multiplication? And if the connection is the Levi Cevita connection, is this saying the structure constants when multiplication is given by $\nabla$ is related to the structure constants when multplication is give by the lie bracket by $\Gamma_{ij}^k-\Gamma_{ji}^k=\gamma_{ij}^k$?

The answer to the first question is not, because the algebra is an infinite-diennsional algebra, while you have only $n^3$ Christoffel symbols. And regarding the second question, if the connection is the Levi-Civita connection then the differences you say are zero, because the Levi-Civita connection is torsion-free.
