I wanted to write a new set of elements, that didn't overlap with those classical four: furthermore, I wanted to encourage the expression of those elements in unexpected ways. However, I wanted the elements to have obvious oppositions, just like the classical set - and to plausibly make up the whole of the world.
I came up with Seasons.
In the Seasons world, dedicate themselves to Spring, Summer, Fall, or Winter. Each mage has only one of these seasons to draw upon, chosen at the time of their dedication.
The Seasons are defined by their 'potentialities.' These are the things that separate them from other seasons. No season is so hot as Summer: no season is so vibrant as Fall. Here's the full list:
Summer:
Heat (Opposes Winter's Cold)
Storm (Opposes Spring's Aspiration)
Virulence (Opposes Fall's Harvest)
Fall:
Endings (Opposes Spring's Beginnings)
Vibrance (Opposes Winter's Absence)
The Harvest (Opposes Summer's Virulence)
Winter:
Cold (Opposes Summer's Heat)
Death (Opposes Spring's Life)
Absence (Opposes Fall's Vibrance)
Spring
Beginnings (Opposes Fall's Endings)
Life (Opposes Winter's Death)
Aspiration (Opposes Summer's Storm)
All the Seasons want to express their potentialities. Summer naturally wants to create heat, chaos, and overflowing life: Winter magic wants to create cold, death, and absence. The Season pushes at the circumstances of its wielders, attracting the things of that season to them and vice versa.
However! While the seasons are broadly defined by their 'potentialities', the mages of a given season draw on the 'foci' of that season. These are the things that, despite belonging to one of the season's opposing potentialities, remain that season's nonetheless. A summer mage, rather than drawing on the great, unfocussed heat of summer, calls upon the cold in summer - the Focus of 'Hail.'
Here's a list of all the Foci:
Summer
Cold-in-Summer (Hail)
Aspiration-in-Summer (Conquest)
Harvest-in-Summer (Wildfire)
Fall
Beginnings-in-Fall (Sunset)
Absence-in-Fall (Departures)
Virulence-in-Fall (Fungus)
Winter
Heat-in-winter (The Hearth)
Life-in-winter (Revolution)
Vibrance-in-Winter (The Aurora)
Spring
Endings-in-Spring (The Thaw)
Death-in-Spring (Stillbirth)
Storm-in-Spring (The Flood)
Thus, every Mantled mage has either chosen a set of tools that are horribly unaesthetic and awkward, or they've chosen a season with goals that they hate. (Or are ambivalent to both.) Most Winter mages will tell you they're in it for the tools: Most Summer ones will tell you they're in it for the shared goal. Neither of them should be taken at face value.
Handy Table