Let G equals U(1) equals the set of complex numbers z with length 1 be the set of unit complex numbers. This is a group: if you multiply two elements of U(1) together you get another unit complex number. Geometrically, it's a circle in the complex plane.

Any z in U(1) can be written as z equals e to the i theta for some \theta. The map i\theta\mapsto e^{i\theta} is going to be our exponential map in this example and it goes from the imaginary numbers i R to U(1). The commutator bracket on i R vanishes: i theta 1 bracket i theta 2 equals minus theta 1 theta 2 plus theta 2 theta 1 equals zero, and the Baker-Campbell-Hausdorff formula reduces to the usual law of logarithms e to the i theta 1 times e to the i theta 2 equals e to the i (theta 1 plus theta 2).

If we translate the line of imaginary numbers so that it passes through the identity element 1 in U(1) then we get a tangent line to the circle U(1). This will be true in general: little g will be (parallel to) the tangent space to G at the identity.