Under what circumstances should the pilot not flying intervene without asking for aircraft control?
There is transfer of control -- "I have the aircraft" / "You have the aircraft."  Then there is intervening -- which may or may not involve a transfer of control... directing "you're getting slow, ADD POWER NOW" may be a more effective example of intervening than taking control, depending on the phase of flight &amp; other circumstances.  (It's easier for a First Officer to intervene with a directive "add power" or "go around" than than for him to take the aircraft!)  And then there is what was shown in the video, which looks like neither of the above -- simply crew duties as @Lnafziger commented. The captain there didn't take over control (yet -- the video ends before he takes over to taxi), they simply both pushed forward on the control column -- put weight on the nosewheel for better steering, perhaps.  Looked perfectly normal &amp; expected.

To answer the question as asked ("Under what circumstances should the pilot not flying intervene without asking for aircraft control?"), there are plenty of times when intervening without taking control could be appropriate -- the directive callout ("ADD POWER!"), or correcting an aircraft system or Flight Management Computer setting, for example.  There are also a smaller set of times when "I have the aircraft!" is appropriate: a disoriented pilot is about to fly into an unusual attitude, or a pilot is incapacitated in a phase of flight where there is no time for "hey, are you okay?" to differentiate between distracted &amp; incapacitated.  The captain always has the prerogative to take over, although there are plenty of times when something else is a better choice -- directing "go around" instead of taking the aircraft at low altitude to try &amp; salvage a bad landing, for instance.  Particularly in cases of incapacitation, the FO also has that option, but realistically, he may have more explaining to do than the captain would have if roles were reversed.

In a pure training situation (initial formation flying in the military comes to mind), there are times when control may transfer back &amp; forth... the student flies as far as he can, starts to get out of position, the instructor takes the aircraft, puts it back into position, the student is given control again, etc.  But that's not really the sort of scenario you'd see in airline operations.  There might be times the pilot-monitoring captain (i.e. he is the pilot not-flying on the FO's leg) might take over the FMC programming rather than talking the FO through the necessary changes -- sometimes it's just better to do it &amp; explain what you're doing rather than talk through each entry and button press.  But that's again not really the same as "I have the aircraft."

With qualified crews in the airline world, it just isn't all that common that a situation arises where one pilot has to assume control outside of the normal transfers (i.e. the normal transfer to the captain on landing if he has the only nosewheel steering tiller).

And, as noted in the comments, every transfer, both expected and unexpected, needs to be verbalized by both pilots: "I have the aircraft."  "You have the aircraft."