  Given sphere preserving (M\"obius) transformations in $n$-dimensional
Euclidean space one can use the Poincar\'e extension to obtain sphere
preserving transformations in a half space of $n+1$ dimensions. The Poincar\'e
extension is usually provided either by an explicit formula or by some
geometric construction. We investigate its algebraic background and describe
all available options. The solution is given in terms of one-parameter
subgroups of M\"obius transformations acting on triples of quadratic forms. To
focus on the concepts, this paper deals with the M\"obius transformations of
the real line only.
