Are you sure you can't even view your sent messages once banned? Not sure that's true, but they were the standard hey x i've replied to you here etc.
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You don't understand the question, hilariousandco is saying that you can view the list of sent messages even if you are banned. He is not talking about sending a message to anyone on the forum. Obviously, you are unlikely to be able to do that.
It does appear that he can't via the screenshot provided, but not allowing people to see their outbox seems a bit weird and it's the first I've heard of it.
That's the first I've heard of it, too. I checked and indeed, outgoing messages cannot be viewed by a banned account. By the way, I tried using the message search and it turns out that the search also works only with incoming messages. That's a rather strange solution.