ABU DHABI—A futuristic 35-story tower where doors swoosh open with the wave of a hand houses a little-known firm long used by this emirate to deploy its oil riches around the world.

When Barclays PLC needed to raise capital in 2008, the Abu Dhabi sovereign-wealth fund, known as IPIC, invested more than $5 billion in the U.K. bank. Through a subsidiary, IPIC acquired holdings in German auto maker Daimler AG and Swiss commodities powerhouse Glencore PLC. It helped finance the ultraluxury New York skyscraper One57, nicknamed...