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When asking the question to draw the most important lewis structure, does that mean to just draw the Lewis Structure that makes its elements most stable (in accordance with addition of double bonding and such) ?
Yes, you do want the most stable Lewis Structure, but you should consider things like fulfilling the octet rule and ensuring that each element's formal charge is as close to 0 as possible. I think double bonds would be related to the sharing of electrons and fulfilling the octet rule.
I think that you should also first you would see which would fulfill the octet rule, then see which has the smallest formal charge and then see where there is more electronegativity to make that lewis structure the most important one.
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