The Lake Theatre is seen in Guy Green's "A Patch of Blue" (MGM, 1965) with Sidney Poitier and Shelley Winters. The report comes from Cinema Treasures contributor John Bosley. He says they're playing a Peter Sellers film and that the Alvarado Theatre is also seen briefly.
Both theatres are in the MacArthur Park area. See the Los Angeles Theatres page on the Lake Theatre for more information and a few photos.
The Alvarado, later known as the Park Theatre, was at 710 S. Alvarado. Still there, but now a down-market retail establishment. See the Wilshire Blvd. Movie Palaces page on the Alvarado Theatre for several vintage photos.
On IMDb: "A Patch of Blue"