Linear Lie group.

Question

Suppose that G is a connected Lie group such that the center of G is trivial.

Question:

Is it true that G is isomorphic (as Lie group) to a closed subgroup of a Linear group GL(n,R) for some natural number n. Or maybe there is an evident counterexample ?

Answer 1

The adjoint representation of $G$ realizes it as a closed subgroup of $\mathrm{GL}(\mathfrak{g})$, where $\mathfrak{g}$ is the Lie algebra of $G$ (in general, the kernel of the adjoint representation consists of elements of $G$ that commute with the connected component of $G$ that contains the identity).