Left-invariant frame of the real Heisenberg group

For multiplication $(x,y,z)(a,b,c)=(x+a,y+b,z+c+xb)$, the left-invariant frame isIts Lie brackets of vector fields satisfy $[E_1,E_2]=E_3$ and the other basis brackets vanish.

The horizontal smooth distribution on the real Heisenberg group is spanned by $E_1={\partial }_x$ and $E_2={\partial }_y+x{\partial }_z$. It is not an involutive distribution because their bracket is ${\partial }_z$, but their brackets span the full tangent space. A horizontal curve satisfies $\dot{x}=\alpha$, $\dot{y}=\beta$, and $\dot{z}=x\beta$.

Every point $(P_x,P_y,P_z)$ can be reached from the identity by a smooth curve tangent to the Heisenberg horizontal distribution. Put $a=2(P_z-P_x{P}_y/2+P_x)$ and chooseTheir integrals give $x(1)=P_x$, $y(1)=P_y$, and $z(1)={\int }_0^{1}x(t)\beta (t)dt=P_x{P}_y/2-P_x+a/2=P_z$. The third coordinate measures the effect of the noncommuting horizontal directions; integrability would instead confine horizontal curves to a two-dimensional leaf.