Splitting up a run containing a joker
Rummikub allows you to break up sets and runs to form new groupings as you wish, as long as all groupings on the board are valid at the end of the turn.  This means all sets and runs must consist of at least 3 tiles and follow the other rules for creating them.  Therefore you can always split a run into two (or more) smaller runs, provided that each of the smaller runs still has at least 3 tiles.

However, it's difficult to answer your question as written because the initial state of the tiles on the board is invalid.  A joker tile must "stand in" for another tile that would be valid in the run, so you can't have "10, J, 11" - there can't be a tile in between the 10 and 11.

If we assume that your initial state is a run of 7, 8, 9, 10, J, 12, then it is legal to pull the 7, 8, and 9 away to form a new run, leaving the 10, J, 12 as a valid run.