 Orientation-preserving affine group of the real line

The real affine group used here is the orientation-preserving subgroup of the affine group on the real line. It acts by $x\mapsto e^{a}x+b$ and has multiplication $(a,b)(a^{\prime },b^{\prime })=(a+a^{\prime },b+e^a{b}^{\prime })$. The positive dilation distinguishes this connected component from the full group, which also contains negative dilations.

Here is a small Lie group where order matters. Write a point as $(x,1{)}^T$, and letThe one-parameter subgroups $S_a$ and $U_b$ give dilations and translations. With matrix multiplication acting rightmost first, $S_a{U}_b$ sends $x$ to $e^{a}x+e^{a}b$, whereas $U_b{S}_a$ sends $x$ to $e^{a}x+b$. At $a=\log 2$ and $b=1$, translate then double gives $2x+2$, while double then translate gives $2x+1$. Starting at $x=1$ gives endpoints $4$ and $3$.

The same order dependence survives infinitesimally: the matrix commutator is $[D,T]=DT-TD=T$. Indeed, the group commutator satisfiesAt $a=b=0.1$, the residual translation is $0.01051709$, close to the leading product $ab=0.01$. It measures a second-order failure of the four moves to cancel. This connects finite composition directly to the Lie bracket on the Lie algebra; see Meinrenken, Section 4, Proposition 4.7.