Past offenses are always off limits during a trial.
Even during sentencing (in states where the jury decides the sentence) a lot is not revealed.
But this was a show trial so San Francisco could show the world they are happy to sacrifice their own citizens on the altar of political correctness. They picked a jury of idiots who would deliver the verdict they wanted. That jury just became accessories to murder.
Past offenses are always off limits during a trial.
Not always. There was a famous trial in Arizona, a self defense case, where the defense wanted to include the past of the man who was shot to show that he was violent, and that made the shooter’s testimony about his actions credible.
It went to the State Appeals Court, which ruled that the past history should have been allowed before the jury.
https://www.law.umich.edu/special/exoneration/Pages/casedetail.aspx?caseid=4266