Looks very solidly BSA indeed, the piled arms logo on the tank rather says that.
The BSA bit is hidden behind the gearchange.
Twin belt rim brakes (one each wheel) makes it early 1920s, a large sidevalve single makes it a Model L (350cc) or Model H (557cc).
Muffler under the engine makes it early 1920s, rather then mid 1920s.
A very well equipped family machine by the look of it, moderately expensive at the time.
Sold in considerable numbers...
This one a 1922 Model H
Couldn't readily find a larger pic, smaller pics seem plentiful - bit strange.
Someone has catalog brochure pics in sequence somewhere, will have a look later.
http://www.vintage-motorcycle.com/index.php?language=en&site=3&pid=1&id=339