I apologize in advance if my question is elementary. Before I present my question I mention my motivation:

# Motivation:

A Lie group is a manifold. At the same time it is a Riemannian manifold equipped with an invariant metric. Moreover its tangent space at the neutral element is a Lie algebra.

On the other hand every manifold can be embedded in some Euclidean space. Furthermore this embedding can be chosen to be an isometric embedding. Moreover every lie algebra can be embedded in an inner Lie algebra, an algebra with an inner Lie bracket $[a,b]=ab-ba$.

In our question, for a given Lie group, we would like to combine all (or parts of) these properties.

# Question:

Let $G$ be a Lie group with neutral element $e$. Is there an algebra structure on some $\mathbb{R}^n$ and a smooth embedding $f:G \to \mathbb{R}^n$ with $f(e)=0$ such that $Df_e$ carries $T_e G$ to a sub Lie algebra of $\mathbb{R}^n$ with the inner Lie structure $[a,b]=ab-ba$?Can we choose such $f:G \to \mathbb{R}^n$ an isometric embedding where $G$ is equipped with an invariant metric and $\mathbb{R}^n$ with its standard metric? Can we choose such an $f$ such that $Df_g$ carry all tangent spaces $T_g G,\;g\in G $ to a Lie Subalgebra of $\mathbb{R}^n$ as a Lie algebra embedding?

# A trivial example:

The embedding of circle in the plane, when we equipe the plane with an arbitrary commutative algebra structure, satisfies all requirements of the question.

i.e., a closed subgroup of some $\mathrm{GL}_m(\mathbb R)$), and this can obviously be done for linear Lie groups; so maybe it's enough to show that this condition is preserved under covers? (Or, if it's false, then a non-linear group like a spin or metaplectic group is the place to look.) $\endgroup$ – LSpice Feb 17 '18 at 22:50