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What has come to be called Geometric Algebra is a school of thought among some physicists who amplify the good use of Clifford algebra in treatments of basic classical mechanics and quantum mechanics.
This is not about deep mathematical properties of Clifford algebras (for instance something like Bott periodicity or K-theory will never be mentioned in “Geometric Algebra”-texts), but is about clean and useful exposition of basic mechanics (classical and quantum). One point being made here is that the traditional textbook emphasis on the special geometry of the Cartesian space $\mathbb{R}^3$ with its exceptional vector product structure is outdated and contra-productive and can be replaced by a more elegant and more universal description in terms of bivector calculus (which canonically embeds into the Clifford algebra).
Geometric Algebra is, naturally, strongest where it comes to the description of rotation and spin.
At other points the strict emphasis of the Clifford algebra is maybe a bit too much.
The existence of the school of thought called Geometric Algebra goes back to
Comprehensive mechanics textbooks written in this style are for instance