Coordinate formula for the Lie bracket

Fix $p\in U$. By [L4], choose a smooth function $\chi :M\to [0,1]$ that is $1$ on a neighbourhood $W$ of $p$ and has support contained in $U$. For each $j$, let ${\tilde{x}}^j$ be the global smooth function that equals $\chi x^j$ on $U$ and $0$ outside $U$. Then ${\tilde{x}}^j=x^j$ on $W$, and because $\chi$ is constant on $W$, one also has $X({\tilde{x}}^j)=X^j$ and $Y({\tilde{x}}^j)=Y^j$ on $W$.

Apply [L1] to ${\tilde{x}}^j$ and use step 1.1. At $p$ this gives $[X,Y]({\tilde{x}}^j)(p)=X(Y^j)(p)-Y(X^j)(p)={\sum }_i{X}^i(p){\mathrm{\partial }}_i{Y}^j(p)-{\sum }_i{Y}^i(p){\mathrm{\partial }}_i{X}^j(p)\mathrm{.}$ Each resulting coefficient is smooth on $U$.

On $U$, the ordinary product rule gives the same coefficient formula when [L1] acts on any smooth test function: the mixed second derivatives cancel by [L3]. Thus the smooth local field with coefficients from step 2.1 acts as the bracket. On overlaps, two such fields agree on all local test functions, so they agree pointwise; the local fields therefore glue uniquely. This proves the formula without selecting a family of charts or local fields.