Lie Algebra

A Lie algebra is the infinitesimal approximation to a Lie group

A Lie algebra is a vector space $\mathfrak{g}$ equipped with a bilinear skew-symmetric map: $$\begin{gathered} \mathfrak{g}\times \mathfrak{g} \\ to\mathfrak{g} \end{gathered}$$ which satisfies the Jacobi identity:

$$\forall x,y,z\in \mathfrak{g}:\left[x,\left[y,z\right]\right]+\left[z,\left[x,y\right]\right]+\left[y,\left[z,x\right]\right]=0$$