That equivalent central extensions determine the same commutator map follows from ^P3 and Correspondence between 2-cocycles and central extensions.

Now let

be an alternating -bilinear map and let . Let

Then is -bilinear and thus a 2-cocycle, amd . By the Correspondence between 2-cocycles and central extensions, there is a central extension of the form above with 2-cocycle and thus commutator map .

Finally let

be a central extension with the same commutator map . Define a -section of so that

Then

for any , so by the Correspondence between 2-cocycles and central extensions these extensions are equivalent.

Note is a group homomorphism for any since

and that is itself a group homomorphism since for

Furthermore, the induced automorphism for any .

Now let be an automorphism such that . It follows that for some function . Noting that

it follows that for some function , and since

it follows .

Now consider a general . Then

for all , so . Conversely, given we consider the central extension

which has the commutator map , and thus from the above correspondence, this extension is equivalent to the original one, giving an automorphism in .