Du Sautoy nilpotent Lie ring for an elliptic curve

Definition
Consider an elliptic curve $$E$$ given by an equation of the form $$y^2 + x^3 - Dx = 0$$ where $$D \in \mathbb{Z}$$. The du Sautoy nilpotent Lie ring corresponding to $$E$$ is the Lie ring $$L(E)$$ with the following presentation:

$$L(E) = \left \langle x_1,x_2,x_3,x_4,x_5,x_6,y_1,y_2,y_3 \mid [x_1,x_4] = Dy_3, [x_1,x_5] = y_1, [x_1,x_6] = y_2, [x_2,x_4] = y_1, [x_2,x_5] = y_3, [x_3,x_4] = y_2, [x_3,x_6] = y_1, \mbox{other Lie brackets between generating set elements all zero} \right \rangle$$

The Lie ring $$L(E)$$ is canonically attached to the elliptic curve $$E$$.

Related notions

 * Du Sautoy nilpotent group for an elliptic curve