Mr. Putin consolidated his control over Russia’s oligarchs, Mr. Tillerson underwent a profound change of outlook. He came to realize that the key to success in Russia, a country deeply important to Exxon’s future, lay in establishing personal relationships with Mr. Putin and his friend and confidant, Igor Sechin, the powerful head of Rosneft, the state oil company.

And as Mr. Tillerson and other oil executives pivoted from the private sector to the state oil company, the criticism that they had directed toward the Kremlin dried up. The payoff for Exxon was immense: a $500 billion joint venture in 2011 to drill for oil on the Arctic shelf and the Black Sea and another huge deal to develop shale oil deposits in Siberia. Those projects were shelved in 2014, after the West imposed sanctions on Russia for Mr. Putin’s actions in Crimea and Ukraine.