What are the main reasons why it was right to hang Louis Riel?

There is one main reason why Louis Riel was executed (and
why it was right for him to be executed).  The reason is the fact that he committed
treason.  It is very hard to argue that Riel did not commit
treason.

Treason is the crime of betraying your own country
or of trying to overthrow its government.  There is no question that Riel did the second
of these things.  The rebellions that he led were aimed at creating an independent
nation that would be made up of lands that were part of Canada.  Going to war against
your own government for this purpose is clearly treason and treason is typically
punishable by death.

The only real reason to have spared
Riel would have been his mental state.  The fact that the jury asked for his life to be
spared would indicate that perhaps he should not have been executed.  However, the jury
did find that he was mentally capable enough to be found guilty.  If he was mentally
capable of understanding what he was doing, then he was clearly guilty of
treason.

So, your best argument for hanging Riel is the
simple fact that he, apparently knowingly, committed treason.  This is a crime that was
punishable by death at the time (and still is punishable by death in many
places).