Lie third theorem ($\mathrm{Lie}(H)\cong \mathfrak{h}$)

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.