Lie algebra is a vector space $latex g$ with a map $latex [\cdot, \cdot]:g\times g \rightarrow g$ such that
- $latex [\cdot,\cdot]$ is bilinear
- $latex [x,x]=0
- Jacobi inequality: $latex [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0,\forall x,y,z$.
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Note that 2) $latex \Rightarrow [x,y]=-[y,x]$ since $latex 0=[x+y,x+y]=[x,x]+[y,y]+[x,y]+[y,x]$.
Note that the converse is true most of time as well since that implies $latex [x,x]+[x,x]=2[x,x]=0$. As long as we can divide by 2 (in the field where $latex 1+1\neq 0$), we have $latex [x,x]=0$.