That information is out-of-date. AB 51 explicitly made lane splitting legal.
The whole history is that, back in the 1960s or 70s, there was a precedent where a motorcyclist successfully fought a ticket by pointing out that lane splitting was necessary because their air-cooled motorcycle would have overheated in the heavy stop-and-go traffic.
For decades, it was this gray area. The California Highway Patrol (who often practice lane splitting themselves) eventually issued guidelines for how to do it safely. Cyclists who stuck to those guidelines were generally left alone, and others sometimes got generic “reckless driving” tickets.
(The guidelines boiled down to: only to it between the #1 and #2 lanes, only when traffic is heavy and slow, and don’t go more than 10 mph faster than the surrounding traffic.)
Early in this century, somebody challenged the CHP guidelines on the grounds that the CHP didn’t have the authority to create new rules of the road, and, even if they did, they didn’t go through the appropriate rule-making process. The CHP pulled down their official guidelines, making things murkier than they’d been in a while.
A study or two came out and said that, lane splitting done wisely, is actually safer than motorcycles participating in the slow-and-go of heavy congestion: cyclists were less likely to be rear-ended and less likely to be overlooked by other drivers making lane changes in dense congestion.
AB 51 made it explicitly legal if done is a safe and reasonable way and authorized the CHP to make safety guidelines. The CHP now has guidelines very similar to the long-standing ones, though there seems to be less emphasis on only doing it in the leftmost lanes. It’s far more common now to be passed simultaneously on two sides, which is awkward because you can’t shift over to make more room.
Around the same time, we also made a rule that motor vehicles have to give bicyclists at least 3 feet of clearance when passing. I have no idea if a motorcycle and “lane share” with a bike.
Only transplants from Ohio or Indiana.