Abelian complex Lie algebra of dimension 2

The 2-dimensional abelian complex Lie algebra is $\mathbb C^{2}$ with its usual vector-space operations and identically zero bracket:

[x,y]=0\qquad\text{for all }x,y.

Morphisms

Every complex-linear map to another abelian complex Lie algebra preserves the bracket. Hence endomorphisms form $M_{2}(\mathbb C)$ and automorphisms are the invertible matrices. A homomorphism from an arbitrary Lie algebra to this object must kill every commutator.