Wikipedia derivation of the relationship between a torque and an angular acceleration is given here.
Could someone help me to see how the following: $$\vec{\tau} = \left(-\sum^n_{i=1}m_i [\Delta r_i]^2\right) \vec{\alpha} + \vec{\omega} \times \left(-\sum^n_{i=1}m_i [\Delta r_i]^2\right) \vec{\omega}$$ is obtained from the following: $$\vec{\tau} = \sum^n_{i=n} (\vec{r}_i - \vec{R}) \times (m_i \vec{a}_i)$$ by using Jacobi identity, please?
My attempt at the derivation is as follows: $$\vec{\tau} = \sum^n_{i=n} (\vec{r}_i - \vec{R}) \times (m_i \vec{a}_i) \\ \hphantom{\vec{\tau}} = \sum^n_{i=n} \vec{\Delta r_i} \times (m_i \vec{a}_i) \\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times \vec{a}_i) \\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times (\vec{\alpha} \times \vec{\Delta r_i} + \vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})))\;\ldots\;\vec{R}\text{ is centroid} \\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times (\vec{\alpha} \times \vec{\Delta r_i}) + (\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})))\;\ldots\text{ cross-product distributivity over addition} \\ $$
Then, I try the Jacobi identity on the second term as follows: $$\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) + \vec{\omega} \times ((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i}) + (\vec{\omega} \times \vec{\Delta r_i}) \times (\vec{\Delta r_i} \times \vec{\omega}) = \vec{0}$$
The last term on the LHS is $\vec{0}$ because $$(\vec{\omega} \times \vec{\Delta r_i}) \times (\vec{\Delta r_i} \times \vec{\omega}) \\ = (\vec{\omega} \times \vec{\Delta r_i}) \times -(\vec{\omega} \times \vec{\Delta r_i}) \\ = -[(\vec{\omega} \times \vec{\Delta r_i}) \times (\vec{\omega} \times \vec{\Delta r_i})] \\ = \vec{0}$$
So, $$\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) + \vec{\omega} \times ((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i}) = \vec{0}$$
That is different from the one stated in the Wikipedia.
However, I try to continue with my own finding as follows: $$\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) + \vec{\omega} \times ((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i}) = \vec{0} \\ \vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) = -[\vec{\omega} \times ((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i})] \\ \vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) = \vec{\omega} \times -((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i}) \\ \vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) = \vec{\omega} \times (\vec{\Delta r_i} \times (\vec{\omega} \times \vec{\Delta r_i}))$$ to have the following: $$\vec{\tau} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times (\vec{\alpha} \times \vec{\Delta r_i}) + (\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i}))) \\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times (\vec{\alpha} \times \vec{\Delta r_i}) + \vec{\omega} \times (\vec{\Delta r_i} \times (\vec{\omega} \times \vec{\Delta r_i}))) \\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i ([\Delta r_i]^2 \vec{\alpha} - (\vec{\Delta r_i} \cdot \vec{\alpha}) \vec{\Delta r_i} + \vec{\omega} \times ([\Delta r_i]^2 \vec{\omega} - (\vec{\Delta r_i} \cdot \vec{\omega}) \vec{\Delta r_i})) \\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i ([\Delta r_i]^2 \vec{\alpha} - (\vec{\Delta r_i} \cdot \vec{\alpha}) \vec{\Delta r_i} + \vec{\omega} \times [\Delta r_i]^2 \vec{\omega} - \vec{\omega} \times (\vec{\Delta r_i} \cdot \vec{\omega}) \vec{\Delta r_i}) \\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i ([\Delta r_i]^2 \vec{\alpha} - (\vec{\Delta r_i} \cdot \vec{\alpha}) \vec{\Delta r_i} + (0) - \vec{\omega} \times (\vec{\Delta r_i} \cdot \vec{\omega}) \vec{\Delta r_i}) \\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i ([\Delta r_i]^2 \vec{\alpha} - (\vec{\Delta r_i} \cdot \vec{\alpha}) \vec{\Delta r_i} - \vec{\omega} \times (\vec{\Delta r_i} \cdot \vec{\omega}) \vec{\Delta r_i})$$
Then, I get stuck there.
If you want to help, please show the complete derivation instead of just hinting here and there unless you have the complete derivation already in your mind.
