# Three dimensional subalgebras of Clifford Algebras

Are there any three dimensional subalgebras of GA(n) where GA(n) is the geometric algebra corresponding to $R^n$? If yes, what about for GA(2)?

Edit: a geometric algebra is a Clifford algebra.

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When $n=2$ and you take a quadratic form of Euclidean signature (i.e. a positive-definite symmetric bilinear form), for instance, the Clifford algebra is isomorphic to $M_2(\mathbb{R})$, and there is the subalgebra consisting of all matrices of the form $$\begin{pmatrix} * & * \\\ 0 & * \end{pmatrix},$$ or anything conjugate to that.
In general, you end up with either a matrix algebra or a direct sum of two matrix algebras, with coefficients in $\mathbb{R},\mathbb{C}$, or $\mathbb{H}$. In any of those cases, you should be able to find lots of three-dimensional subalgebras.