  The problem of conditioning on the occupation field was investigated for the
Brownian motion in 1998 independently by Aldous [4] and Warren and Yor [34] and
recently for the loop soup at intensity $1/2$ by Werner [35], Sabot and
Tarr\`es [30], and Lupu, Sabot and Tarr\`es [22]. We consider this problem in
the case of the Brownian loop soup on the real line, and show that it is
connected with a flow version of Jacobi processes, called Jacobi flow. We give
a pathwise construction of this flow simultaneously for all parameters by means
of a common Brownian motion, via the perturbed reflecting Brownian motion. The
Jacobi flow is related to Fleming--Viot processes, as established by Bertoin
and Le Gall [9] and Dawson and Li [11]. This relation allows us to interpret
Perkins' disintegration theorem between Feller continuous state
branching-processes and Fleming--Viot processes as a decomposition of Gaussian
measures. Our approach gives a unified framework for the problems of
disintegrating on the real line. The connection with Bass--Burdzy flows which
was drawn in Warren [33] and Lupu, Sabot and Tarr\`es [23] is shown to be valid
in the general case.
