Two sources come to mind, though there may be more recent ones.   These are both available online now.

1) A concise exposition of central extensions is given by Steinberg in Section 7 of  his 1967-68 Yale lectures on Chevalley groups, distributed in mimeographed form and currently linked on his UCLA homepage <a href="http://www.math.ucla.edu/~rst/">here</a>.   Item (ix) at the bottom of page 76 is what you need.   Originally Steinberg developed these ideas mainly in order to justify lifting of projective representations of finite Chevalley groups to such groups of "universal" type.   But along the way he worked out in considerable generality the algebraic analogues of topological connectedness and simple connectedness: the former imitated by a group being perfect (equal to its derived group) and the latter by a group being a *universal central extension* of a given group (possible only when the given group is perfect).

2) An even more concise account can be found in the important 1968 paper by Calvin Moore <a href="http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1968__35__5_0">here</a>.  See his statement in the middle of page 10.
As indicated by the title *Group extensions of $p$-adic and adelic linear groups*, Moore came to these ideas from a rather different direction.   But his approach and Steinberg's soon merged in the work of Matsumoto and others on the Congruence Subgroup Property in the case of Chevalley (split) groups over number fields.