Confusion about relation between inertial an non-inertial reference frames in respect tto the motion of a rigid body

In "Analytical Mechanics" by N. A. Lemos, in page 99 the author determines the time derivative relation between an inertial frame $Sigma$ an an non-inertial frame $Sigma’$ fixed in a rigid body with angular velocity $mathbf{omega}$ around its origin $O$, such that $$ left(frac{d}{d t}right)_{text {inertial}}=left(frac{d}{d t}right)_{text {body}}+omega times$$ Also in this book, on page 100, the author is trying to prove the uniqueness of the angular velocity of he body, and he considers two frames $Sigma$ and $Sigma’$, the latter with angular velocity $omega _1$, such that an arbitrary point $P$ of the body can be represented by the vector $mathbf{r}$ and also be represented by the sum of vectors $mathbf{r_1}$ and $mathbf{R}$, where $mathbf{R}$ is the $Sigma’$ origin position with respect to $Sigma$ and $mathbf{r_1}$ is the point $P$ position with respect to $Sigma’$‘s origin, such that $mathbf{r} = mathbf{R} + mathbf{r_1}$. The author states that

$$ left(frac{d mathbf{r}}{d t}right)_{Sigma}=left(frac{d mathbf{R}}{d t}right)_{Sigma}+left(frac{d mathbf{r_1}}{d t}right)_{Sigma}=left(frac{d mathbf{R}}{d t}right)_{Sigma}+omega_{1} times mathbf{r_1}$$

which is obviously correct in my conception, but when I try to apply the time derivative relation for non inertial systems, I obtain

$$left(frac{d mathbf{r}}{d t}right)_{Sigma}=left(frac{d( mathbf{R} + mathbf{r_1})}{d t}right)_{Sigma’}+ mathbf{omega_1} times (mathbf{R} + mathbf{r_1}) = left(frac{d mathbf{mathbf{R}}}{d t}right)_{Sigma’} + omega_1 times (mathbf{mathbf{R} + mathbf{r_1}}) $$

which is clearly different from the last equation. Where is my mistake?