I'm certain that I was the first to collect Elgin Gulls and the information about them. I always wanted one as a kid and finally got my first Elgin Gull in the 1970s. I still have it today in mint condition with the original red Troxel saddle and "ELGIN" ornament on the Torrington Bevelock handlebar stem. Over the years I have owned a few other Gulls. I also managed to save the original factory 8 x 10 glossy photos and company letters about the Gull when it was first being planned and constructed.
BACK IN THE 1970s I also managed to find an engineer who originally worked on the Gull for Sears and Westfield. This was when nobody cared and there was no hobby, no internet and no "experts." I saved this stuff in the 1970s and I still have all of these items today.
While people in this crazy hobby today are claiming there are only four or five known, this can't be possible. Certainly not if I count up the ones I know about and the ones I have owned. So. While there may be any number of "experts" out there all ready to argue about the Gull. I have been collecting these and saving their history for something like 60 years.
RE: the Gull frame in your photos... Your Gull has simply fallen victim to weathering and a condition known as "age hardening" that takes place with old aluminum– especially from the 1930s. Stuff gets brittle and crumbly and weak. It appears your frame has separated at the assembly points, rather than merely breaking. But this said, I would say your frame is very repairable– at least to get it aesthetically back to how it ought to be. I just would not ever consider attempting to ride it. These bicycles were not good to ride even when new.
And these bicycles were not put together like Silver Kings (I collect these too since childhood)... not at all. So don't go there. Completely different assembly method and design.
Get someone who has expertise in working with aluminum and doing machining work. Get the lug pegs extended with fresh billet back where they should be... and reconnect the tubes. Then have them flush-pinned as original. Then yes, admire it as a display item and leave it at that.
Your handlebar assembly has been replaced. The original factory handlebar was all aluminum. My keeper Gull still has the original all-aluminum assembly intact and in nearly perfect condition. The Gull in the photo appears to have rusted chrome-plated steel handlebars. This tells me right away that the original assembly likely broke as most did and was replaced (as most were) with a steel unit. This was Sears' remedy for broken Gull handlebars (Westfield's remedy too).
I can also see that your saddle stem appears to be either broken off or missing. Again... these were originally made of aluminum billet. And yes, they broke off regularly and when this happened were replaced with steel.
I am attaching a photo of my keeper Gull with some of my other bicycles on display at "Balloonatic '88"... a meet I put on in Southern California, guess when? Please forgive the typo on the photo caption regarding "Sachnoffsky" which should be "Sakhnoffsky." (This is the bicycle that today's "experts" call "pod bike"....got this one in the 1970s too).
For what it is worth.
National Bicycle History Archive of America (NBHAA.com)