Similar to the derivation of separation of angular momentum into $L_{CM}$ and $L_{internal}$, one can derive similar expression for Energy as $E = \frac{1}{2}M_{total}v_{CM}^{2} + \frac{1}{2}\sum \mu_{i} v_{i}^{'2}$.

Proof: $$E = \frac{1}{2}\sum \mu_{i} v_{i}^{2}$$ $$v_{i} = v_{CM} + v_{i}^{'}$$ $$E = \frac{1}{2}\sum \mu_{i} v_{CM}^{2} + v_{CM}\sum \mu_{i} v_{i} + \frac{1}{2}\sum \mu_{i} v_{i}^{'2}$$ Since in CoM frame $\sum \mu_{i} r_{i} =0 \to \sum \mu_{i} v_{i}=0$. $$QED$$

L and E within Com frame can be related only if body is rigid. One can refer Klepner & Kolenkow Classical Mechanics.

Similar to the derivation of separation of angular momentum into $L_{CM}$ and $L_{internal}$, one can derive similar expression for Energy as $E = \frac{1}{2}M_{total}v_{CM}^{2} + \frac{1}{2}\sum \mu_{i} v_{i}^{'2}$.

Proof: $$E = \frac{1}{2}\sum \mu_{i} v_{i}^{2}$$ $$v_{i} = v_{CM} + v_{i}^{'}$$ $$E = \frac{1}{2}\sum \mu_{i} v_{CM}^{2} + v_{CM}\sum \mu_{i} v_{i}^{'} + \frac{1}{2}\sum \mu_{i} v_{i}^{'2}$$ Since in CoM frame $\sum \mu_{i} (r_{i}-r_\text{cm}) =0 \to \sum \mu_{i} v_{i}^{'}=0$. $$QED$$

L and E within Com frame can be related only if body is rigid. One can refer Klepner & Kolenkow Classical Mechanics.