Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$ and $r \in \bigwedge^2 \mathfrak{g}$. For $x \in G$ let $\lambda_x$ denote the left multiplication by $x$. Let $[\cdot, \cdot]$ denote the Schouten bracket for multivector fields. Then show that $$\left [\left ( d_e \lambda_x \right )^{\otimes 2} (r), \left (d_e \lambda_x \right )^{\otimes 2} (r) \right ] = \left (d_e \lambda_x \right )^{\otimes 3} \left ([[r,r]] \right )$$ where $[[r,r]] = \left [r_{12}, r_{13} \right ] + \left [r_{12}, r_{23} \right ] + \left [r_{13}, r_{23} \right ]$, i.e., if $r = \sum_i x_i \otimes y_i$, then $$[[r,r]] = \sum_{i,j} [x_i, x_j] \otimes y_i \otimes y_j + \sum_{i,j} x_i \otimes [y_i, x_j] \otimes y_j + \sum_{i,j} x_i \otimes x_j \otimes [y_i, y_j] \,.$$
This amounts to say that the left action respects the Schouten bracket. I have asked the same question in MSE and here's the link but unfortunately the question is closed. This is the place I got stuck and hence I am seeking help. Could anyone give some suggestion or insight as to how to come up with this equality?
Thanks for your time.
Source $:$ Etingof lecture notes on compact quantum groups (Alternative intuitive approach to Theorem $3.1$ in page no. $28$).
