We study the Grassmann geometry of surfaces when the ambient space is a 3-dimensional unimodular Lie group with left invariant metric, that is, it is one of the 3-dimensional commutative Lie group, the 3-dimensional Heisenberg group, the groups of rigid motions on the Euclidean or the Minkowski planes, the special unitary group $SU(2)$, and the special real linear group $SL(2,\mathbb R)$.
"Grassmann geometry on the 3-dimensional unimodular Lie groups I." Hokkaido Math. J. 38 (3) 427 - 496, August 2009. https://doi.org/10.14492/hokmj/1258553972