Twitter Inc. said it would appoint Tesla Inc. Chief Executive Elon Musk to its board, essentially bringing the billionaire social-media agitator inside the fold.

Mr. Musk on Monday disclosed a 9.2% stake in the company, making him its largest shareholder. He has been one of its louder critics, assailing the platform on free-speech grounds. His frequent, colorful use of Twitter has also landed him in occasional legal trouble.

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