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If you are VERY careful with their placement then french cleats would work well at supporting that with the added bonus (assuming you only want to have a shelf for support) that you could dispense with the supporting shelf. One fixed to the bottom rail of the frame to support the base and one fixed to each upright about 2/3 of the way up.
Alternatively these floating shelf brackets are good. Easy to install and strong.
Bobbo-
I think the shelf will complete the frame visually and I don’t trust the bottom rail, which I think is a later addition (presumably after it was torn off a fireplace)!
A French cleat seems to be the best way, it’s just lining them up that is tricky. -
I have called french cleats 'split battens' for years and had no idea about the correct name!
Dramatic_Hammer
fredtc
Couple of pictures for context. The frame is quite thick but possibly made of (antique) cheese - the whole thing seems to flex when moved.