Sorry for a belated reply.
Yes, of course, this can be done. First you merge the lines (GeomMergeLines), then you split them at the intersections and remove duplicate parts (GeomNormalize), then you output each branch as a separate geom (GeomToBranches).
CREATE TABLE [splitlines_t] (
INDEX [mfd_id_x] BTREE ([mfd_id]),
INDEX [geom_x] RTREE ([geom]),
PROPERTY 'FieldCoordSystem.geom' ComponentCoordSystem([closedlines])
CREATE DRAWING [splitlines] (
PROPERTY 'Table' '[splitlines_t]',
PROPERTY 'FieldGeom' 'geom'
INSERT INTO [splitlines_t] ([geom])
SELECT [value] FROM (
SELECT SPLIT CALL GeomToBranches(GeomNormalize(GeomMergeLines([Geom]), 0))
Here's how the result looks like:
I made the existing lines thick gray, kept the split lines thin black and put them on top. Then I At-clicked one of the split lines to show where it starts and ends.