Tenon lengths can vary depending on the needs of the frame. Normally some standard rules are that the tenon thickness is 1/4 the timber width.
For example an 8" timber has a 2" thick tenon and it is usually offset from the reference face the same distance. Making it a 2" offset.
So, for a 6x6 it would be 1 1/2" tenon and its mortise and offset 1 1/2" as well.
Now for peg sizes it is usually 1/2 the tenon thickness. So for a two inch tenon you'd use a 1" peg. And for a 1 1/2" tenon a 3/4" peg.
I make my brace tenons 3" long, usually.

Good luck with your project.

Jim Rogers
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Whatever you do, have fun doing it!