A Grassmann number is a member of an exterior algebra. There are two important operations on Grassmann numbers: the Hodge star conjugate and the dagger conjugate (parity). If these operations are involutions, then one can decompose a Grassmann number in terms of a self-conjugate and an anti-self-conjugate part.
1-Grassmann Numbers
A 1-Grassmann number has the form
$$A = a_{0} + a_{1} W.$$
This variable has two pieces of information.
Star Conjugate and Star Decomposition
The Hodge star conjugate is
$$ A^{\star} = a_{1} + a_{0}W. $$
Note that the star conjugate is an involution:
$$ (A^{\star})^{\star} = A. $$
The star decomposition of \(A\) is
$$ A = \frac{1}{2} \left( A + A^{\star} \right) + \frac{1}{2} \left( A - A^{\star} \right). $$
The first term is the self-star-conjugate part:
$$ \frac{1}{2} \left( A + A^{\star} \right) = \frac{1}{2} \left( a_{0} + a_{1} \right) \left( 1 + W \right). $$
The second term is the anti-self-star-conjugate part:
$$ \frac{1}{2} \left( A - A^{\star} \right) = \frac{1}{2} \left( a_{0} - a_{1} \right) \left( 1 - W \right). $$
Each of these parts has one piece of information about \(A\).
Dagger Conjugate and Dagger Decomposition
The dagger conjugate is
$$ A^{\dagger} = a_{0} - a_{1}W. $$
Note that the dagger conjugate is also an involution:
$$ (A^{\dagger})^{\dagger} = A. $$
The dagger decomposition of \(A\) is
$$ A = \frac{1}{2} \left( A + A^{\dagger} \right) + \frac{1}{2} \left( A - A^{\dagger} \right). $$
The first term is the self-dagger-conjugate part:
$$ \frac{1}{2} \left( A + A^{\dagger} \right) = a_{0}. $$
The second term is the anti-self-dagger-conjugate part:
$$ \frac{1}{2} \left( A - A^{\dagger} \right) = a_{1}W. $$
Each of these parts has one piece of information about \(A\).
Note that the star and dagger conjugates do not commute:
$$ (A^{\star})^{\dagger} = a_{1} - a_{0} W, \qquad (A^{\dagger})^{\star} = -a_{1} + a_{0} W = - (A^{\star})^{\dagger}. $$
2-Grassmann Numbers
A 2-Grassmann number has the form
$$ A = a_{1} + a_{W}W + a_{X}X + a_{WX}WX. $$
Note that a 2-Grassmann number has four pieces of information.
Star and Dagger
The star conjugate is
$$ A^{\star} = a_{WX} - a_{X}W + a_{W}X + a_{1}WX. $$
Note that the star conjugate is not an involution:
$$ (A^{\star})^{\star} = a_{1} - a_{W}W - a_{X}X + a_{WX}WX. $$
Indeed, the double star conjugate is equivalent to the dagger conjugate:
$$ A^{\dagger} = a_{1} - a_{W}W - a_{X}X + a_{WX}WX. $$
The dagger conjugate is an involution:
$$ (A^{\dagger}) = A. $$
The dagger decomposition of \(A\) is
$$ A = \frac{1}{2} \left( A + A^{\dagger} \right) + \frac{1}{2} \left( A - A^{\dagger} \right). $$
The first term is the self-dagger-conjugate part:
$$ \frac{1}{2} \left( A + A^{\dagger} \right) = a_{1} + a_{WX}WX. $$
The second term is the anti-self-dagger-conjugate part:
$$ \frac{1}{2} \left( A - A^{\dagger} \right) = a_{W}W + a_{X}X. $$
Each part has two pieces of information about \(A\).
Note that the star conjugate is an involution on the self-dagger-conjugate part:
$$((a_{1} + a_{WX}WX)^{\star})^{\star} = (a_{WX} + a_{1}WX)^{\star} = a_{1} + a_{WX}WX. $$
Thus, the self-dagger-conjugate part can be decomposed into self-star-conjugate and anti-self-star-conjugate parts:
$$ a_{1} + a_{WX}WX = \frac{1}{2} \left( a_{1} + a_{WX} \right) \left( 1 + WX \right) + \frac{1}{2} \left( a_{1} - a_{WX} \right) \left( 1 - WX \right). $$
On the other hand, the star conjugate is an anti-involution on the anti-self-dagger-conjugate part:
$$ ((a_{W}W + a_{X}X)^{\star})^{\star} = (-a_{X}W + a_{W}X)^{\star} = -(a_{W}W + a_{X}X). $$
Indeed, the eigenvalues of the 2-dimensional star conjugate operation are \(\lambda = \pm 1\) and \(\lambda \pm i\). All four satisfy \(\lambda^{4} = 1\). The star decomposition is as follows:
$$ A = w + x + y + z; $$
where
$$ w^{\star} = w, \qquad x^{\star} = -x, \qquad y^{\star} = iy, \qquad z^{\star} = -iz. $$
It follows that
$$ w = \frac{1}{4} \left( A + A^{\star} + A^{\dagger} + (A^{\dagger})^{\star} \right); $$
$$ x = \frac{1}{4} \left( A - A^{\star} + A^{\dagger} - (A^{\dagger})^{\star} \right); $$
$$ y = \frac{1}{4} \left( A - i A^{\star} - A^{\dagger} + i (A^{\dagger})^{\star} \right); $$
$$ z = \frac{1}{4} \left( A + i A^{\star} - A^{\dagger} - i (A^{\dagger})^{\star} \right). $$
3-Grassmann Numbers
A 3-Grassmann number has the form
$$ A = a_{1} + a_{W}W + a_{X}X + a_{WX}WX + a_{Y}Y + a_{WY}WY + a_{XY}XY + a_{WXY}(WX)Y. $$
Note that a 3-Grassmann number has eight pieces of information.
Star Conjugate and Star Decomposition
The star conjugate is
$$ A^{\star} = a_{WXY} + a_{XY}W - a_{WY}X + a_{Y}WX + a_{WX}Y - a_{X}WY + a_{W}XY + a_{1}(WX)Y. $$
Note that this is an involution. Thus, \(A\) can be star-decomposed:
$$ A = \frac{1}{2} \left( A + A^{\star} \right) + \frac{1}{2} \left( A - A^{\star} \right). $$
The self-star-conjugate part of \(A\) is
$$ A + A^{\star} = (a_{1} + a_{WXY}) (1 + (WX)Y) + (a_{W} + a_{XY})(W + XY) + (a_{X} - a_{WY})(X - WY) + (a_{Y} + a_{WX})(Y + WX). $$
The anti-self-star-conjugate part of \(A\) is
$$ A - A^{\star} = (a_{1} - a_{WXY}) (1 - (WX)Y) + (a_{W} - a_{XY})(W - XY) + (a_{X} + a_{WY})(X + WY) + (a_{Y} - a_{WX})(Y - WX). $$
Each part has four pieces of information.
Dagger Conjugate and Dagger Decomposition
The dagger conjugate is
$$ A^{\dagger} = a_{1} - a_{W}W - a_{X}X + a_{WX}WX - a_{Y}Y + a_{WY}WY + a_{XY}XY - a_{WXY}(WX)Y. $$
Note that this is an involution. Thus, \(A\) can be dagger-decomposed:
$$ A = \frac{1}{2} \left( A + A^{\dagger} \right) + \frac{1}{2} \left( A - A^{\dagger} \right). $$
The self-dagger-conjugate part of \(A\) is
$$ \frac{1}{2} (A + A^{\dagger}) = a_{1} + a_{WX}WX + a_{WY}WY + a_{XY}XY. $$
The anti-self-dagger-conjugate part of \(A\) is
$$ \frac{1}{2} (A - A^{\dagger}) = a_{W}W + a_{X}X + a_{Y}Y + a_{WXY}(WX)Y. $$
Note that the dagger conjugate of the self-star-conjugate part is anti-self-star-conjugate:
$$ (A + A^{\star})^{\dagger} = (a_{1} + a_{WXY}) (1 - (WX)Y) - (a_{W} + a_{XY})(W - XY) - (a_{X} - a_{WY})(X + WY) - (a_{Y} + a_{WX})(Y - WX); $$
and similarly, the dagger conjugate of the anti-self-star-conjugate part is self-star-conjugate:
$$ (A - A^{\star})^{\dagger} = (a_{1} - a_{WXY}) (1 + (WX)Y) - (a_{W} - a_{XY})(W + XY) - (a_{X} + a_{WY})(X - WY) - (a_{Y} - a_{WX})(Y + WX). $$
Each part has four pieces of information.
4-Grassmann Numbers
A member of the 4-Grassmann algebra has the form
$$ A = A_{0} + A_{1} + A_{2} + A_{3} + A_{4}; $$
where
$$ A_{0} = a_{1}; $$
$$ A_{1} = a_{W}W + a_{X}X + a_{Y}Y + a_{Z}Z; $$
$$ A_{2} = a_{WX}WX + a_{WY}WY + a_{WZ}WZ + a_{XY}XY + a_{XZ}XZ + a_{YZ}YZ; $$
$$ A_{3} = a_{WXY}(WX)Y + a_{WXZ}(WX)Z + a_{WYZ}(WY)Z + a_{XYZ}(XY)Z; $$
$$ A_{4} = a_{WXYZ} ((WX)Y)Z. $$
Note that a 4-Grassmann number has sixteen pieces of information.
Star and Dagger
The star conjugate is
$$ A^{\star} = (A_{0})^{\star} + (A_{1})^{\star} + (A_{2})^{\star} + (A_{3})^{\star} + (A_{4})^{\star}; $$
where
$$ (A_{0})^{\star} = a_{1}((WX)Y)Z; $$
$$ (A_{1})^{\star} = a_{W}(XY)Z - a_{X}(WY)Z + a_{Y}(WX)Z - a_{Z}(WX)Y; $$
$$ (A_{2})^{\star} = a_{WX}YZ - a_{WY}XZ + a_{WZ}XY + a_{XY}WZ - a_{XZ}WY + a_{YZ}WX; $$
$$ (A_{3})^{\star} = a_{WXY}Z - a_{WXZ}Y + a_{WYZ}X - a_{XYZ}W; $$
$$ (A_{4})^{\star} = a_{WXYZ}. $$
Note that this operation is not an involution. Just as for 2-Grassmann numbers, you find that
$$ ((A_{0})^{\star})^{\star} = A_{0}, \qquad ((A_{2})^{\star})^{\star} = A_{2}, \qquad ((A_{4})^{\star})^{\star} = A_{4}. $$
However, you also have
$$ ((A_{1})^{\star})^{\star} = -A_{1}, \qquad ((A_{3})^{\star})^{\star} = -A_{3}. $$
Again, the double star conjugate is equivalent to the dagger conjugate:
$$ A^{\dagger} = (A^{\star})^{\star}. $$
The dagger conjugate is an involution. The dagger conjugate is
$$ A^{\dagger} = (A_{0})^{\dagger} + (A_{1})^{\dagger} + (A_{2})^{\dagger} + (A_{3})^{\dagger} + (A_{4})^{\dagger}; $$
where
$$ (A_{0})^{\dagger} = +A_{0}; $$
$$ (A_{1})^{\dagger} = -A_{1}; $$
$$ (A_{2})^{\dagger} = +A_{2}; $$
$$ (A_{3})^{\dagger} = -A_{3}; $$
$$ (A_{4})^{\dagger} = +A_{4}. $$
This means that on \(A_{0}\), \(A_{2}\), and \(A_{4}\) the star conjugate is an involution. Thus you can perform a star decomposition:
$$ A_{0} + A_{2} + A_{4} = \frac{1}{2} \left( A_{0} + A_{2} + A_{4} + (A_{0})^{\star} + (A_{2})^{\star} + (A_{4})^{\star} \right) + \frac{1}{2} \left( A_{0} + A_{2} + A_{4} - (A_{0})^{\star} - (A_{2})^{\star} - (A_{4})^{\star} \right). $$
The self-star-conjugate part is
$$ \left(a_{1} + a_{WXYZ} \right) \left(1 + ((WX)Y)Z \right) + \left(a_{WX} + a_{YZ}\right)\left(WX + YZ \right) + \left(a_{WY} - a_{XZ}\right)\left(WY - XZ \right) + \left(a_{WZ} + a_{XY}\right)\left(WZ + XY \right). $$
The anti-self-star-conjugate part is
$$ \left(a_{1} - a_{WXYZ} \right) \left(1 - ((WX)Y)Z \right) + \left(a_{WX} - a_{YZ}\right)\left(WX - YZ \right) + \left(a_{WY} + a_{XZ}\right)\left(WY + XZ \right) + \left(a_{WZ} - a_{XY}\right)\left(WZ - XY \right). $$
