How to do $\frac{d}{d q} \|q^*_c \times q - q_I\|^2$ (quaternion)

Question

How to do the following $$\frac{d}{d q} \|q^*_c \times q - q_I\|^2,$$ where

1. $q$ : a unit quaternion, i.e., $q^Tq=1$
2. $\times$ : vector cross product
3. $q_I$ : $[0 \ \ 0 \ \ 0 \ \ 1]^T$, i.e., no rotation
4. $q^*_c$ : a constant unit quaternion

My effort is that it becomes

$$2(q^*_c \times q -q_I) \frac{d}{d q}(q^*_c \times q)=2(q^*_r \times q -q_I) q^*_c\times I_{4}.$$

The last part is from the following

Derivative of cross product wrt vector

It looks like the dimension is not correct, how to modify it?

And how to move onto the next step?

Thanks!