The moment of truth has come for Elon Musk and Tesla.

The Securities and Exchange Commission sued the company’s chief executive, chairman and largest shareholder for securities fraud on Thursday. The complaint alleges that Mr. Musk claimed falsely on social media that funding was “secured” for a Tesla buyout yet “knew that he had never discussed a going-private transaction at $420 [a share] with any potential funding source.” Tesla announced it had decided to remain public less than three weeks later.

Both...