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