In an interview the SEC commissioner showed that he wasn't even aware of the tweet by Musk. He asked during the interview, something like "he said that ?!".
Either way, not sure what they can do. Infact I think it adds to the pressure on them to act on short sellers. Transparency is totally lacking, as we all know. Nasdaq has prposed some transparency rules after these events (may be they were working on these before we don't know). High time SEC did something.
The biggest problem, probably said already here, is high frequency trading. Congress needs to enact a small tax on it with revenues accruing to SEC for watchdog keyed to performance.