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Let F be a separably closed field of characteristic different from 2 and 3. We consider 3-dimensional subspaces V of trace zero cube-central order 3 matrices with coefficients in F . Each such vector space gives rise to a ternary cubic form, mapping an element of V to its cube. We use geometric properties of cubic curves to determine V up to conjugacy in the case where the cubic form is nonsingular.

Classifying subspaces of cube-central order 3 matrices using the geometry of cubic curves