 Euclidean motion Lie algebra in two dimensions

The real Lie algebra $\mathfrak{iso}(2)$ is a semidirect product of planar translations with infinitesimal rotations. One orientation convention is $[J,E_1]=-E_2$, $[J,E_2]=E_1$, $[E_1,E_2]=0$. Its translation space is a two-dimensional abelian ideal of a Lie algebra.