Problem: Given $X, Y$ vector fields on manifold $M$ and regarded as sections of tangent bundle of $M$, i.e., $X, Y: M \rightarrow TM$, how do we define their composition $X \circ Y$? For example, if we consider $X: A \rightarrow (A, AR), Y: A \rightarrow (A, RA)$ on $O(n)$ with $R$ a skew symmetric matrix, then certainly $X, Y: O(n) \rightarrow TO(n)$, then what is $X \circ Y$? It seems that if we consider $X \circ Y(A)$, $Y$ is already mapping $A$ to $TO(n)$, so what is $X$ doing to the resulting element in $TO(n)$? Any help or reference is appreciated.
Edit: Ok, I believe one way for me to think of the vector fields as derivations in the example I provided above is to find the flows of the vector fields, and differentiate with respect to time.
Assuming your manifold is a smooth manifold and $X,Y:M\to TM$ are vector fields that are also smooth. Then let $p\in M$ and $\textbf{x}:U\subseteq \mathbb{R}^n\to M$ be a parameterization. Since $X,Y$ are smooth there exists smooth functions $(a_k(p))$ and $(b_k(p))$ on $U$ such that $$X(p)=\sum_ka_k(p)\frac{\partial}{\partial x_k}, Y=\sum_kb_k(p)\frac{\partial}{\partial x_k}$$ where $\{\partial/\partial x_k\}$ is the basis associated with the parameterization $\textbf{x}$. Let $f$ be a differentiable function on the manifold $M$. Then $X\circ Y$ is defined as $$(X\circ Y)(f)(p):=X(Yf)(p)=X((\sum_kb_k\frac{\partial}{\partial x_k})f)(p)=X(\sum_kb_k\frac{\partial f}{\partial x_k})(p)\\=(\sum_ka_k\frac{\partial}{\partial x_k})(\sum_kb_k\frac{\partial f}{\partial x_k})(p)=(\sum_{i,j}a_i\frac{\partial b_j}{\partial x_i}\frac{\partial f}{\partial x_j}+\sum_{i,j}a_ib_j\frac{\partial^2 f}{\partial x_ix_j})(p)$$