I know that the Euclidean group $SE(3)$ in the $3$ dimensions has the Lie algebra structure as the semidirect product of skew symmetric matrices and $\mathbb{R}^3$.
However, I cannot find a text saying anything about the simplicity or at least semisimplicity of this Lie algebra.
If it is semisimple, I can argue that it is not compact and so the Yang-Mills gauge lagrangian for $SE(3)$ is not positive-definite.
Could anyone help me with the simplicity of the Lie algebra of $SE(3)$?
By $\mathbb R^3$ you mean an abelian $3$-dimensional Lie algebra which furthermore (as exhibited by the semidirect product) is an ideal of the Lie algebra in question. But the only nonzero ideals of semisimple Lie algebras are simple Lie algebras, and simple Lie algebras are not abelian.