A quick newspapers.com search found reference to the company in San Diego, CA, in 1914. Thence to the San Diego directory for that year, see attached. I have not checked adjacent years, nor looked for additional info. Happy to research further if you need (ie want).
It seems that the "Silver Gate" was the entrance to San Diego harbor, just as the "Golden Gate " was to San Francisco - news to me, and interesting.
Silver Gate Soda Works, San Diego Directory, 1914.jpg
Yes, the Silver Gate was San Diego. In the 19th century, LA really wasn't much to talk about. San Diego and San Francisco were the big cities.
I dug this Silver Gate soda from the late 19th early 20th century Federal Avenue city dump in the 60s (I was about 12). The bottle is a blown-in-mold, applied lip crown top produced probably about 1906 or so shortly after the crown top was invented. I have a number beer and sodas made in that manner from that dump.