English translation:
Well then
In that case
Structure:
では + Phrase
それでは + Phrase
じゃあ + Phrase
Could there be notes added to the grammar page detailing the differences between では・それでは・じゃあ? I’m guessing it’s just a matter of formality, but it would be good to have confirmation on the page.
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Hey and sorry for the late answer 
You are right! The difference comes to formality.
それでは and its contraction では have the same level of formality and are used in formal writing, formal/polite speech.
In casual language, それじゃあ、それじゃ、じゃあ and じゃ are used instead.
I will add this information to the grammar point!
Thank you
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Still not all that clear on the grammar page on the different levels. It lists those like you say, but I had the following:
______、ここで失礼します
それじゃ was listed as too casual, but it accepted じゃあ
That’s because じゃあ is a tad more formal than それじゃ！
Just came to this point and it brought back some memories.
I’m an ALT. A homeroom teacher I worked with a couple of years ago had her own unique way of saying this (at least I’ve never heard anyone else say it):
「それではんじゃあ」
She must’ve said it at least half a dozen times every lesson and I and the students would lightly tease her about it (all in good fun). It definitely rubbed off though and I find myself saying it occasionally 