It is known since [7] that any group definable in an o-minimal expansion of the real field can be equipped with a Lie group structure. It is then natural to ask when does a Lie group is Lie isomorphic to a group definable in such expansion. Conversano, Starchenko and the first author answered this question in [2] in the case where the group is solvable. We give here a criterion in the case where the group is linear. More precisely if G is a linear Lie group it is isomorphic to a group definable in an o-minimal expansion of the reals if and only if its solvable radical is isomorphic to such group.

We characterize, up to Lie isomorphism, the real Lie groups that are definable in an o-minimal expansion of the real field. For any such group, we find a Lie-isomorphic group definable in R exp \mathbb {R}_{\exp } for which any Lie automorphism is definable.

We characterize, up to Lie isomorphism, the real Lie groups that are definable in an o-minimal expansion of the real field.

It has been known since (Pillay, J. Pure Appl. Algebra 53 (1988), no. 3, 239-255)that any group definable in an 𝑜-minimal expansion of the real field can be equipped with a Lie group structure. It is therefore natural to ask when is a Lie group Lie isomorphic to a group definable in such an expansion. Conversano, Starchenko and the first author answered this question in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441-452) in the case when the group is solvable. This paper answers similar questions in more general contexts. We first give a complete classification in the case when the group is linear. Specifically, a linear Lie group 𝐺 is Lie isomorphic to a group definable in an 𝑜-minimal expansion of the reals if and only if its solvable radical has the same property. We then deal with the general case of a connected Lie group, although unfortunately, we cannot achieve a full characterization. Assuming that a Lie group 𝐺 has its Levi subgroups with finite center, we prove that in order for 𝐺 to be Lie isomorphic to a definable group it is necessary and sufficient that its solvable radical satisfies the conditions given in (Conversano, Onshuus,