In case one of the engines fails after taking off (let's suppose the left engine is the one that fails), which direction should we turn to go back to the airport? A left or a right traffic pattern?
That depends on a lot of factors. Are you declaring an emergency? What altitude are you at what speed are you at when this happens etc? What kind of aircraft are you flying? Does it have enough power to continue to climb on one engine? Some trainers do not.
For a typical airliner, loss of a single engine is not a big deal. The pilot will simply declare an emergency and get vectors from controllers as to where to go land again. Most airliners have plenty of power with one engine out and they don't have to worry about that. Smaller twin engined GA aircraft are a different story and may not be able to achieve a positive rate of climb on one engine, especially with the gear and flaps extended just after liftoff. At low altitudes where this performance factor may be the case for the specific airplane you are flying, you may have to execute a forced landing in a clearing either directly ahead or off to your side, as a 180 deg turn to the airfield would be suicide.
As to maneuvering once an engine is lost, there is no requirement for this, save a pre-determined VFR traffic patterns for the departure airport.
If the engine out is an emergency and you cannot comply with either a controller's instructions or uncontrolled procedures, you are authorized as PIC to disregard any regulation as required to deal with the emergency. This means any kind of maneuvering you feel will safely bring the aircraft back to the ground is acceptable.
I've read a lot of posts and comments amongst pilots about not turning into a dead engine on a twin; this is nonsense with one caveat - you must have sufficient airspeed in order to do this. It depends on the aircraft in question but a good rule of thumb is that if you are at or above Vyse you can GENTLY turn into a dead engine. The threat comes in attempting to turn into the dead engine at or near Vmc, minimum controllable airspeed on a single engine, where it may not be possible to arrest the turn due to thrust asymmetry.
- Don't turn into the dead engine, turn away from it, you have to keep the dead engine in front of the live one otherwise loss of control can occur at low speeds.
- I assume you can maintain height / have sufficient altitude and are at or above blue line speed if you are considering turning because some height may be lost in the turn. Some twins can only just about maintain height on one engine depending on aircraft loading etc. If you are low / only just at blue line you should seriously consider closing the throttle on the good engine and trying to make the best of a bad job or extend out and see if you can gain some height first before doing anything rash. Hopefully extending out might allow you to find a suitable place to put the aircraft down. This is far preferable to stalling and spinning trying to make the runway at low speed / low height.