Every finite-dimensional real Lie algebra is the Lie algebra of a connected simply connected Lie group. One construction first uses the Ado theorem to embed it into a matrix Lie algebra, integrates the corresponding left-invariant distribution by the Frobenius theorem to a connected immersed Lie subgroup, and then takes a universal covering Lie group. The immersed subgroup need not be closed; taking its ambient closure may change its Lie algebra.
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