Killing frame for a right-invariant Heisenberg metric (${\partial }_x,{\partial }_y+x{\partial }_z,{\partial }_z$)

Every right-invariant Riemannian metric on the Heisenberg group has Killing vector fields ${\partial }_x$, ${\partial }_y+x{\partial }_z$ and ${\partial }_z$. Their Lie brackets of vector fields reproduce $[X,Y]=Z$ and the central relations. Their flows are right translations. The dual frame of the right-invariant coframe is different and has the opposite Lie-bracket sign.