  This work is developed in the context of Lorentzian spin-foams with space-
and time-like boundaries. It is argued that the equations describing the
semiclassical regime of the various spin-foam amplitudes admit a common
biquaternionic structure. A correspondence is given between Majorana 2-spinors
and time-like surfaces in Minkowski 3-space based on such complexified
quaternions. A symplectic structure for Majorana spinors is constructed, with
which the unitary representation theory of $\mathrm{SU}(1, 1)$ is re-derived.
As the main result, we propose a symplectomorphism between Majorana spinor
space (with an area constraint) and $T^*\mathrm{SU}(1, 1)$, generalizing
previous studies on twisted geometries to the case of time-like 2-surfaces.
