where "formula_7" indicates terms involving higher commutators of formula_4 and formula_5. If formula_4 and formula_5 are sufficiently small elements of the Lie algebra formula_12 of a Lie group formula_13, the series is convergent. Meanwhile, every element formula_14 sufficiently close to the identity in formula_13 can be expressed as formula_16 for a small formula_4 in formula_12. Thus, we can say that "near the identity" the group multiplication in formula_13—written as formula_20—can be expressed in purely Lie algebraic terms. The Baker–Campbell–Hausdorff formula can be used to give comparatively simple proofs of deep results in the Lie group–Lie algebra correspondence.