From the book "Spinning Tops" by Audin, she claims that
$$\mathfrak{so}(3)[\epsilon]/\epsilon^2$$ with coefficientwise Lie bracket is a Lie algebra of a Lie group that is $TSO(3)$ (group action not specified) or of the group of rigid motions in $\mathbb{R}^3$, that is, in either case, the semidirect product of $SO(3)$ and $\mathbb{R}^3$ given by standard $SO(3)$ action on $\mathbb{R}^3$.
Is this group the $E(3)$ of euclidean group of $\mathbb{R}^3$? And how do I extract such basis $\{1,\epsilon\}$ as above?