= Solution
In a coordinate chart centred at the identity, smooth multiplication has a <local group law>
$$
z^k=F^k(y,x)=x^k+y^k+B^k{}_{ij}y^ix^j+O(3).
$$
The identity gives $F(0,x)=x$ and $F(y,0)=y$, while <associativity> gives the functional equation $F(u,F(y,x))=F(F(u,y),x)$. Smooth inversion determines coordinates $\bar x$ with $F(\bar x,x)=F(x,\bar x)=0$. Changes of local coordinates alter the symmetric part of $B^k{}_{ij}$, whereas its antisymmetric part supplies the coordinate-independent <Lie bracket> on the tangent space at the identity.
Back to article page
