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We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an ominimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = R · S where R is the solvable radical of G. We also prove that any semisimple subalgebra of the Lie algebra of G corresponds to a unique ind-definable semisimple subgroup of G.doi:10.4064/fm222-1-3 fatcat:mbuwsgvwerd25o2vqgpmvgwk6q