  In this article, we extend Siegel's cross-ratio identity for $2\times2$
determinants over the truncated polynomial ring
$F[\varepsilon]_\nu:=F[\varepsilon]/\varepsilon^\nu$. We compute cross-ratios
and Goncharov's triple-ratios in $F[\varepsilon]_2$ and $F[\varepsilon]_3$ and
use them extensively in our computations for the tangential complexes. We also
verify a "projected five-term" relation in the group $T\mathcal{B}_2(F)$ which
is crucial to prove one of our central statements that describe the morphisms
between tangent complex and Grassmannian complex
