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Let $\mathcal{B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra ofdiagonal type corresponding to a matrix $\mathfrak{q} \in \mathbf{k}^{\theta\times \theta}$, where $\mathbf{k}$ is an algebraically closed field ofcharacteristic 0. Let $\mathcal{L}_{\mathfrak{q}}$ be the Lusztig algebraassociated to $\mathcal{B}_{\mathfrak{q}}$, seehttp://arxiv.org/abs/1501.04518. We present $\mathcal{L}_{\mathfrak{q}}$ as anextension (as braided Hopf algebras) of $\mathcal{B}_{\mathfrak{q}}$ by$\mathfrak Z_{\mathfrak{q}}$ where $\mathfrak Z_{\mathfrak{q}}$ is isomorphicto the universal enveloping algebra of a Lie algebra $\mathfrakn_{\mathfrak{q}}$. We compute the Lie algebra $\mathfrak n_{\mathfrak{q}}$ when$\theta = 2$.

A finite-dimensional Lie algebra arising from a Nichols algebra of diagonal type (rank 2)