Lie Algebra

For a vector space $V$ defined on field $F$ , a Lie algebra is tuple $⟨V,[\cdot ,\cdot ]⟩$ , where $[\cdot ,\cdot ]$ is a bilinear mapping: $[\cdot ,\cdot ]:V\times V\to V$ , s.t. all of the 3 following conditions are fulfilled:

1. Bilinearity:

$$[ax+by,z]=a[x,z]+b[y,z],\mathrm{\forall }x,y,z\in V.$$

2. Anti-Symmetry:

$$[x,x]=0,\mathrm{\forall }x\in V.$$

If the field $F$ does not have character $2$ , we also have

$$[x,y]=-[y,x],\mathrm{\forall }x,y\in V.$$

3. Jacobi Identity:

$$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0,\mathrm{\forall }x,y,z\in V.$$