just to include my admiration as well - a wonderful piece showing art nouveau decoration at the height of its fame.
if I may add a little to the sound advice given so far …...………… the gilding is indeed ultra thin, so perhaps not wise to do any rubbing other than a very light and dry dust, and you will need to live with whatever appearance the gold now has. Obviously gold doesn't oxidise in a real sense, but can show a tarnished appearance if exposed to air pollutants such as open fires, tobacco smoke and cooking fumes etc., most of which, fortunately, are issues we no longer live with. Looking at the top edges of the text block of older books, where the edges have been gilded, will show the kind of damage that gilding can suffer when not protected from such pollutants, and the top edges of books kept without protection become dull and dirty, and sadly this deterioration can't be reversed.
Although not mentioned by the others, my suggestion would be that this piece needs to be protected, within a fully enclosed cabinet etc., which hopefully will avoid any more deterioration to the gilding.
From a purely practical perspective, I would also add - if you don't have facilities to provide the sort of care and protection that this piece needs, and sentiment isn't necessarily high on your list, then selling this vase might be a viable option. Of course if the piece does have massive family value then just pretend I didn't offer this suggestion.

P.S. appreciate I'm a little out of touch with modernity, but did think I knew most of the commonly used acronyms - but I seem to be ignorant of 'fab' - is it some sort of praiseworthy expression?