Matrix Representation and Lie Algebra basis of a product of Lie Groups * Is there a principled way to find the matrix Lie group (representation) and Lie Algebra basis of the product of multiple Lie groups? In particular, and as an example, say $G = \mathrm{SE(3)} \times \mathbb{R}^3 \times \mathrm{SE(3)}$.

For the Lie algebra, Ado's theorem gives a faithful linear representation $\mathfrak{g}\rightarrow \mathfrak{gl}_n(K)$ for some $n$. Of course, if $G=G_1\times G_2$ is already linear, then we can represent $\mathfrak{g}=\mathfrak{g}_1\oplus \mathfrak{g_2}$ by block matrices, and take the natural representation for the linear Lie algebras $\mathfrak{g}_1,\mathfrak{g}_2$. In the example here, $SE(3)$ is already a linear Lie group. A basis for its Lie algebra is given here.