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Since the geometric product of two collinear vectors is a scalar, the symmetric part of the geometric product is a scalar denoted . The product a · b is the inner or scalar product. One can then decompose the geometric product into its symmetric and antisymmetric part:

where the antisymmetric part is formed with the outer product. For the outer product one has obviously a b = −b a and a a = 0, so that a b can be interpreted geometrically as an oriented area. The geometric product is constructed in such a way that it gives information over the relative directions of a and b, i.e., ab = ba = a · b a b = 0 means that a and b are collinear whereas ab = −ba = a b a · b = 0 means that a and b are perpendicular.

With the outer product one defines simple r-vectors or r-blades

which can be interpreted as r-dimensional volume forms. The geometric product can then be generalized to the case of a vector and an r-blade:

which is the sum of an (r−1)-blade and a (r + 1)-blade . Applying this recursively one sees that each c-number can be written as a polynomial of r-blades, and using a set of basis vectors e₁, e₂, … , e_r a c-number reads:

A is called a multivector or r-vector if the highest appearing grade is r. It decomposes into several blades:

where _n projects onto the term of grade n. A multivector A_r is called homogeneous if all appearing blades have the same grade, i.e., A_r = A_{r r}. The geometric product of two homogeneous multivectors A_r and B_s can be written as

The inner and the outer product stand now for the terms with the lowest and the highest grade:

One should note that the inner and outer product here in the general case do not correspond anymore to the symmetric and the antisymmetric part of the geometric product. For example, in the case of two bivectors one has A₂ B₂ = B₂ A₂, so that the outer product is symmetric. Actually one finds for the symmetric and the antisymmetric parts of A₂B₂:

In general the commutativity of the outer and the inner product is given by:

and both products are always distributive:

Only the outer product of r-vectors is in general associative, i.e., A (B C) = (A B) C, for the inner product one gets:

If one has to calculate several products of different type, the inner and the outer product always have to be calculated first, i.e.,

The formalism of geometric algebra briefly sketched so far can now be described with Grassmann variables and the Clifford star product, that turns the Grassmann algebra into a Clifford algebra. To make the equivalence even more obvious we go over to the dimensionless Grassmann variables

These variables play here the role of dimensionless basis vectors and will therefore be written in bold face, whereas the θ_i played in the discussion of the first section the role of dynamical variables with dimension . In the σ_n-variables the Clifford star product (2.12) has the form

As a star product the Clifford star product is associative and distributive.

To show how the geometric algebra described with Grassmann variables and the Clifford star product looks like, we first consider the two-dimensional euclidian case. One has then two Grassmann basis elements σ₁ and σ₂, so that a general element of the Clifford algebra is a supernumber A = a₀ + a₁σ₁ + a₂σ₂ + a₁₂σ₁ σ₂ = A ₀ + A ₁ + A ₂ and a vector corresponds to a supernumber with Grassmann grade one: a = a₁σ₁ + a₂σ₂. The Clifford star product of two of these supernumbers is

where the symmetric and the antisymmetric part of the Clifford star product is given by:

and

which are terms with Grassmann grade 0 and 2, respectively. Note that now a juxtaposition like ab is just as in the notation of superanalysis the product of supernumbers and not the Clifford product, which we want to describe explicitly with the star product (3.14). The σ_i form an orthogonal basis under the scalar product: .

The unit 2-blade i = σ₁σ₂ can be interpreted as the generator of -rotations because by multiplying from the right one gets

so that a vector x = x₁σ₁ + x₂σ₂ is transformed into . The relation describes then a reflection and furthermore one has with (2.16): , so that i corresponds to the imaginary unit. The connection between the two-dimensional vector space with vectors x and the Gauss plane with complex numbers z is established by star multiplying x with σ₁:

Such a bivector that results from star multiplying two vectors is also called spinor. While the bivector i generates a rotation of when acting from the right, the spinor z generates a general combination of a rotation and dilation when acting from the right. One can see this by writing with . Acting from the right with z causes then a dilation by |z| and a rotation by , one has for example: σ_{1 C}z=x, which is the inversion of (3.19). Here one can see that the formalism of geometric algebra reproduces complex analysis and gives it a geometric meaning.

After having described the geometric algebra of the euclidian 2-space we now turn to the euclidian 3-space with basis vectors σ₁, σ₂, and σ₃ and with the Clifford star product (3.14) for d = 3. The basis vectors are orthogonal: σ_i · σ_j = δ_ij and a general c-number written as a supernumber has the form

This multivector has now four different simple multivector parts. Besides the scalar part a₀ there is the pseudoscalar part corresponding to I₃ = σ₁σ₂σ₃, which can be interpreted as a right handed volume form, because a parity operation gives (−σ₁) (−σ₂) (−σ₃) = −I₃. Moreover I₃ has also the properties of an imaginary unit: and I_{3 C}I₃ = I₃ · I₃ = −1. While the pseudoscalar I₃ is an oriented volume element the bivector part with the basic 2-blades

describes oriented area elements. Each of the i_r plays in the plane it defines the same role as the i of the two-dimensional euclidian plane defined above. Star-multiplying with the pseudoscalar I₃ is equivalent to taking the Hodge dual, for example to each bivector B = b₁i₁ + b₂i₂ + b₃i₃ corresponds a vector b = b₁σ₁ + b₂σ₂ + b₃σ₃, which can be expressed by the equation B = I_{3 C}b. This duality can for example be used to write the geometric product of two vectors a = a₁σ₁ + a₂σ₂ + a₃σ₃ and b = b₁σ₁ + b₂σ₂ + b₃σ₃ as:

where and a×b=ε^klma_kb_lσ_m. Furthermore one finds:

and cyclic permutations. Note also that one gets with the nabla operator _x=σ₁∂_x1+σ₂∂_x2+σ₃∂_x3 for the gradient of a vector field f = f₁ (x₁, x₂, x₃) σ₁ + f₂ (x₁, x₂, x₃) σ₂ + f₃ (x₁, x₂, x₃) σ₃:

The multivector part of (3.20) with even Grassmann grade have the basis 1, i₁, i₂, i₃ and form a closed subalgebra under the Clifford star product, namely the quaternion algebra. The multivector part of (3.20) with odd grade does not close under the Clifford star product, but nevertheless one can reinvestigate the definition of the Pauli functions in (2.14). Replacing in (2.14) the scalar i by the pseudoscalar I₃ one sees that the basis vectors σ_i fulfill

which justifies denoting them σ_i. With the pseudoscalar I₃ the trace (2.18) can be written as Tr (F) = 2∫dσ₃ dσ₂dσ₁ F = 2∫dσ₃ dσ₂ dσ₁I_{3 C}F. So one has here achieved with the Clifford star product a cliffordization of the three-dimensional Grassmann algebra of the σ_i.

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