What does cos(wt) says to e(j) ? Get real

When dealing everyday with baseband complex signals, I often forget those are not real (yes).
I write this post in order to commit strongly to not forgetting those lines.

Constant envelope baseband signal - complex

$$\begin{aligned} s_b(t) &= e^{j\theta(t)} \cr & = \cos\theta(t)+j\sin\theta(t) \cr & = i + jq \end{aligned}$$

Send it - real

$$\begin{aligned} s_u(t) &= 2\Re [e^{j\theta(t)}e^{j \omega t}] \cr & = 2\Re [(i + jq)(\cos\omega t + j \sin \omega t)] \cr & = 2(i\cos \omega t - q\sin \omega t) \cr \end{aligned}$$

Get it back - complex

$$\begin{aligned} s_d(t) &= s_u(t)\cos\omega t - j s_u(t)\sin \omega t \cr & = 2 (i\cos^2\omega t+j\sin^2\omega t - q\sin \omega t\cos \omega t - ji\sin \omega t\cos \omega t)\cr & = 2\left(\frac{i}{2}+j\frac{q}{2}+\frac{\cos(2\omega t)}{2}-j\frac{\cos(2\omega t)}{2}- q\sin \omega t\cos \omega t - ji\sin \omega t\cos \omega t\right)\cr & \text{... Apply Low Pass Filter} \cr \mathbf{}{s_d(t)} &= s_b(t) \end{aligned}$$