Following Bernard Looney’s demise, just who is in the running to take over at BP?

BP Plc’s board woke up on Wednesday facing an urgent search for a new boss, its fifth in 16 years. Outgoing Chief Executive Officer Bernard Looney shocked the industry a day earlier with his resignation after failing to fully disclose past relationships with colleagues.

The storied oil and gas producer has typically molded CEOs over decades-long careers within the company. But with Looney becoming the third of the last four chiefs to leave BP abruptly, ...