Suppose is a Lie group. and are right and left translations. Assume is a base of , the tangent space at the unit element . denote the right-invariant vector field obtained by the right translation and by left translation. Let and denote the dual of and . Then we have the Maurer-Cartan equation

Here are constants. Also we have the structure equation with respect to . We see that there is a relation between and , that is

Choose a local coordinate system at . Then

where . Apply exterior differentiation on both sides

Letting , we get

Then it is easy to see .

see Chern’s *lectures on differential geometry*. p182

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