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In geometric algebra, a simple k-blade may be defined as one which can be written as the outer product of $k$ vectors. For example, a 2-blade $A$ is one which may be factored as $A=\mathbf{a}\wedge\mathbf{b}$. This is still a bivector, but not all bivectors are 2-blades in $n$-dimensions. For example, in 4-dimensional space the bivector $\mathbf{e}_1\wedge\mathbf{e}_2 + \mathbf{e}_3\wedge\mathbf{e}_4$ cannot be written as the product of two blades; it is not simple.

Books like Geometric Algebra for Computer Science interpret $k$-blades as $k$-dimensional homogenous subspaces of the vector space in which the geometric algebra is defined. However, they also claim that "we have no geometric interpretation for such nonblades in our vector space model" (Section 2.9.3, p. 46). They prove that the above non-simple bivector does not correspond to any subspace of $\mathbb{R}^4$, however. Therefore, they conclude that "the geometric role of elements, if any, is different" (ibid.).

Hence, there seems to be trouble expanding the geometric significance of $k$-bladesto $k$-vectors generally. Is there any interpretation elsewhere in the literature? There are certainly references for the algebraic properties of $k$-vectors and multi-vectors, but the geometric significance seems to be relatively underdeveloped.

Because elements of the geometric algebra may be interpreted as operators (e.g. rotors, versors, etc), perhaps they may be interpreted in this way as well?

Or perhaps the issue comes with defining addition in the algebra. In the algebra of matrices, we can add them, but the results are harder to interpret than multiplication. For example, we can compose two rotations by multiplying their orthogonal matrices, but what geometric significance does the sum of these matrices hold?

Cheers

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