Jaime Tran, 28, allegedly carried out the attacks on Feb. 15 and 16, according to federal prosecutors.
"He was on the radar, if you will, of our FBI partners in what we call a 'Guardian lead' because of hate-fueled remarks or emails that he was sending to former classmates of a dental school which he was attending," Moore said at a Los Angeles Police Commission meeting.
Both victims wore clothing that identified their faith, including black coats and head coverings, authorities said. Tran, arrested on the evening of Feb. 16, told law enforcement that he looked online for a "kosher market" and decided to shoot someone nearby, according at an affidavit filed by the FBI. He also admitted to shooting someone the previous day, the affidavit said.
Tran had been the "subject of mental health evaluations and was and has been suffering from mental illness through the years," Moore said.
"How he came into the possession of that rifle as well as the handgun, given his mental health condition, is the subject of our continued investigation," Moore said.
Tran was arrested in the Riverside County community of Cathedral City near Palm Springs.
Prosecutors say suspect looked up kosher markets in LA to target Jewish people for shootings
According to the affidavit, Los Angeles police officers investigating the second shooting used video from a camera at an intersection to identify an older model gray Honda that appeared to be involved.
An officer who responded to assist saw and photographed a man driving a dark gray Honda Civic. The image captured the license plate, which was registered to Tran, whose driver license photo was consistent with witness' descriptions of the shooter, the affidavit said.
License plate readers showed the Honda was in the area of the two shootings at the times they occurred. Police identified a mobile phone number associated with Tran and location data showed it was in the Palm Springs area on the afternoon of Feb. 16.
Around 5:45 p.m., Cathedral City police responded to a call from someone who heard the sound of a gunshot and saw a man with a gun near a Honda Civic.
Officers found Tran standing next to the car, and they could see an "AK-style rifle" and a .380-caliber handgun in plain view on the driver's seat, the affidavit said. The officers also found a spent shell casing.
"How he came into the possession of that rifle as well as the handgun, given his mental health condition, is the subject of our continued investigation," Moore said. "Was everything done? The FBI is continuing to evaluate and go back over their efforts these last few years in regards to interplay with him and ensuring everything was done that could be done."
The Associated Press contributed to this report.