For any Lie group G, if there exists a Lie group G' and a surjective homomorphism with an Abelian Lie group as its kernel, such that there does not exist any right inverse (i.e. a homomorphism such that αβ is the identity morphism), then we say G' is a central extension of G.
Examples
We can now give it a central extension into the Lie algebra spanned by E', P'i, C'i, L'ij (antisymmetric tensor), M such that M commutes with everything (i.e. lies in the center, that's why it's called a central extension) and