  We investigate observability and Lipschitz stability for the Heisenberg heat
equation on the rectangular domain $$\Omega =
(-1,1)\times\mathbb{T}\times\mathbb{T}$$ taking as observation regions slices
of the form $\omega=(a,b) \times \mathbb{T} \times \mathbb{T}$ or tubes $\omega
= (a,b) \times \omega_y \times \mathbb{T}$, with $-1<a<b<1$. We prove that
observability fails for an arbitrary time $T>0$ but both observability and
Lipschitz stability hold true after a positive minimal time, which depends on
the distance between $\omega$ and the boundary of $\Omega$: $$T_{\min}
\geqslant \frac{1}{8} \min\{(1+a)^2,(1-b)^2\}.$$ Our proof follows a mixed
strategy which combines the approach by Lebeau and Robbiano, which relies on
Fourier decomposition, with Carleman inequalities for the heat equations that
are solved by the Fourier modes. We extend the analysis to the unbounded domain
$(-1,1)\times\mathbb{T}\times\mathbb{R}$.
