Clifford-like parallelisms

Given two parallelisms of a projective space we describe a construction, called blending, that yields a (possibly new) parallelism of this space. For a projective double space (P,‖ℓ,‖r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathbb P},{\mathrel {\parallel _{\ell }}},{\mathrel {\parallel _{r}}})$$\end{document} over a quaternion skew field we characterise the “Clifford-like” parallelisms, i.e. the blends of the Clifford parallelisms ‖ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrel {\parallel _{\ell }}$$\end{document} and ‖r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrel {\parallel _{r}}$$\end{document}, in a geometric and an algebraic way. Finally, we establish necessary and sufficient conditions for the existence of Clifford-like parallelisms that are not Clifford.


Introduction
The first definition of parallel lines in the real projective 3-space dates back to 1873 and was introduced by W.K. Clifford in the metric framework of elliptic geometry (see [5]): two distinct lines M and N in the real elliptic 3-space, are said to be Clifford parallel, if the four lines M, N, M ⊥ and N ⊥ are elements of the same regulus.(Here M ⊥ denotes the polar line of M w.r.t. the "absolute", i.e. the imaginary quadric that determines the elliptic metric in the real projective 3-space).
Some years later, in 1890 F. Klein revived Clifford's ideas and, using the complexification of the real projective space, defined two lines to be parallel in the sense of Clifford if they meet the same complex conjugate pair of generators of the absolute (see [21]).
Depending on the kind of generators under consideration, one can speak of right parallel, or left parallel lines, then each fixed conjugate pair of generators "indicates" a left (or right) parallel class, which in fact is a regular spread, namely an elliptic linear congruence of the projective space.
Thus we can say that a Clifford parallelism in the real projective 3-space consists of all regular spreads, or elliptic linear congruences, whose indicator lines are pairs of complex conjugate lines of a regulus contained in an imaginary quadric.Besides, Clifford parallelisms go in pairs, and also note that all (real) Clifford parallelisms are projectively equivalent.An interesting survey on the various definitions of Clifford parallelisms can be found in [1].
Generalising this situation, H. Karzel, H.-J. Kroll and K. Sörensen in 1973 introduced the notion of a projective double space (P, ℓ , r ), that is a projective space (of unspecified dimension, over an unspecified field) equipped with two parallelism relations fulfilling a configurational property which can be expressed by the axiom (DS) of Section 3 (see [17], [18]).The real projective 3-space with left and right Clifford parallelisms is an example and it turns out that the projective double spaces (P, ℓ , r ) with ℓ r are necessarily of dimension 3 and precisely the ones that can be obtained from a quaternion skew field H over a field F as in Section 4 (see [17], [18], [15], [22]).
In this way one obtains what in 2010 A. Blunck, S. Pasotti and S. Pianta called generalized Clifford parallelisms in the note [4].If the maximal commutative subfields of H are not mutually F-isomorphic, then new "non-Clifford" regular parallelisms can be obtained by "blending" in some suitable way the left and right parallel classes (see [4, 4.13]).This method has no equivalent in the classical case, since the maximal commutative subfields of the real quaternions are mutually Risomorphic.
Taking up this idea, we introduce here the definition of Clifford-like parallelism in a projective double space (P, ℓ , r ), that is a parallelism on P such that In Section 2 we start from the more general setting of equivalence relations on a set L and we define a blend of two equivalence relations π 1 , π 2 as an equivalence relation π 3 such that each equivalence class of π 3 coincides with an equivalence class of π 1 or π 2 .In order to obtain a characterisation of all the blends of π 1 and π 2 in Proposition 2.4, we use the equivalence relation π 12 generated by them, which is the join of π 1 and π 2 in the lattice of equivalence relations on L.
In Section 3 we study the blends of parallelisms of a projective space.By Theorem 3.1, we can prove that the Clifford-like parallelisms of a projective double space (P, ℓ , r ) are precisely the "blends" of ℓ and r .Therefore Clifford-like parallelisms are regular.
In Section 4 we connect a projective double space (P, ℓ , r ) with ℓ r to a quaternion skew field H over a field F and we describe the equivalence relation π ℓr generated by ℓ and r using the maximal commutative subfields of H (see Theorem 4.2).In Theorem 4.10 we obtain a characterisation of all Clifford-like parallelisms of P(H), ℓ , r showing that they are precisely those introduced in [4, 4.13].Finally, in Theorems 4.12 and 4.15 we discuss the existence and some properties of Clifford-like parallelisms that are not Clifford.
To conclude, we observe that it might be interesting to investigate the blends of the left and right parallelisms of an arbitrary kinematic space in the same spirit as in [24].

Blends of equivalence relations
Throughout this section we consider an arbitrary set L. Let π ⊆ L × L be an equivalence relation on L. The partition of L associated with π is denoted by Π.The elements of Π are called π-classes.For any M ∈ L we denote by C(M) the πclass containing M. The same kind of notation will be used for other equivalence relations on L by writing, for example, π 1 , Π 1 and C 1 (M).
The following simple lemma will be used repeatedly.We now introduce our basic notion.
Definition 2.2.Let π 1 and π 2 be (not necessarily distinct) equivalence relations on L. An equivalence relation π 3 on L is called a blend of π 1 and π 2 if Equivalently, condition (2.1) can be written in the form The trivial blends of π 1 and π 2 are the relations π 1 and π 2 themselves.Our aim is to describe all blends of equivalence relations π 1 and π 2 on L. We thereby use that all equivalence relations on L constitute a lattice; see, for example, [2, Ch.I, §8, Ex. 9] or [27,Sect. 50].In this lattice, the meet of π 1 and π 2 equals π 1 ∩ π 2 (however, the meet of equivalence relations is irrelevant for our investigation).The join of π 1 and π 2 is the intersection of all equivalence relations on L that contain π 1 ∪ π 2 or, in other words, the equivalence relation generated by π 1 and π 2 .This join is denoted by π 12 .For all M, N ∈ L, we have M π 12 N precisely when there exist an integer n ≥ 1 and (not necessarily distinct) elements (2.3) Also, we need another elementary lemma.(2.4) Then the set is a partition of L, whose associated equivalence relation π B is a blend of π 1 and π 2 .
(b) Conversely, any blend of π 1 and π 2 arises according to (a) from at least one subset of L.
(c) Let D and D be subsets of L. Applying the construction from (a) to D and D gives π B and π B, respectively.Then π B coincides with π B if, and only if, where Proof.(a) We read off from (2.4) and Lemma 2.1 that B admits a partition by π 12classes.Now Lemma 2.3 shows that B admits a partition by π 1 -classes, namely , and also a partition by π 2 -classes.Applying Lemma 2.1 to the latter, gives the existence of a partition of a partition of L, and so π B is a blend of π 1 and π 2 .
(b) Given any blend π 3 of π 1 and π 2 we start by defining According to its definition, B ′ admits a partition Σ ′ by π 1 -classes.From (2.7), Σ ′ is a partition of B ′ by π 3 -classes as well.Now Lemma 2.1 gives that L \ B ′ admits a partition, say Σ ′′ , by π 3 -classes.No element of Σ ′′ can be in Π 1 .Since π 3 is a blend of π 1 and π 2 , we obtain Σ ′′ ⊆ Π 2 .Next, by virtue of Lemma 2.1, B ′ admits also a partition by π 2 -classes and, finally, Lemma 2.3 provides a partition Σ ′′′ of B ′ by π 12 -classes.We now proceed as in part (a) of the current proof, commencing with the set D given in (2.7).From Lemma 2.1 and due to the existence of the partition Σ ′′′ of B ′ , we see that the set B from (2.4) equals the set B ′ appearing in (2.7).Under these circumstances we end up with (c) This is an immediate consequence of (2.5).
The set D from Proposition 2.4 (a) merely serves the purpose of defining the set B in (2.4).Formula (2.6) does not impose any restriction on D ∩ L 12 and D ∩ L 12 .Therefore, whenever L 12 is non-empty, there is a choice of D and D such that π B = π B even though B B. For example, D := ∅ and D := L 12 ∅ give rise to ∅ = B B, whereas The final result in this section will lead us to a characterisation of blends of parallelisms in Theorem 3.1.It is motivated by the following evident observation.Let π 1 , π 2 , π 3 be equivalence relations on L. If π 3 is a blend of π 1 and π 2 then, by In our current setting, (2.8) is not sufficient for π 3 to be a blend of π 1 and π 2 .Take, for example, as L any set with at least two elements, let π 1 = π 2 = L × L, and let π 3 be the equality relation on L. Then (2.8) is trivially true, but π 3 fails to be a blend of π 1 and π 2 .
Proposition 2.5.Let π 1 , π 2 , π 3 be equivalence relations on L such that (2.8) is satisfied.Then (2.9) Proof.Assume, to the contrary, that (2.9) does not hold.So, there is an Likewise, C 3 (M 0 ) C 2 (M 0 ) implies that there exists an element Thus, by (2.8), at least one of the following is satisfied.

Blends of parallelisms
We consider a projective space P with point set P and line set L. An equivalence relation on L is called a parallelism on P if each point q ∈ P is incident with precisely one line from each equivalence class; see, for example, [12], [13], or [16, § 14].The notation from the previous section will slightly be altered when dealing with parallelisms by writing instead of π.In addition, if ⊆ L × L is a parallelism, then the equivalence class of a line M ∈ L will be called its parallel class, and it will be denoted by S(M) in order to emphasise the fact that S(M) is a spread of P. On the other hand, the partition of L arising from will be written as Π like before.In the presence of several parallelisms we shall distinguish between these objects by adding appropriate indices or attributes.
As anticipated, the next theorem provides a characterisation of blends of parallelisms by virtue of Proposition 2.5.
Theorem 3.1.Let 1 and 2 be parallelisms on P. Then the following hold.
(a) Any blend of 1 and 2 is a parallelism on P.
(b) A parallelism 3 on P is a blend of 1 and 2 if, and only if, Proof.(a) All parallel classes of the given parallelisms are spreads of P. The same applies therefore to all equivalence classes of any blend of 1 and 2 , that is, such a blend is a parallelism on P. (b) If 3 is a blend of 1 and 2 then (3.1) is nothing but a reformulation of (2.8).Conversely, we first make use of Proposition 2.5, which gives (2.9) up to some notational differences.Next, we notice that no proper subset of a spread of P is again a spread of P. Since all parallel classes of 1 , 2 , and 3 are spreads of P, we are therefore in a position to infer from (2.9) that, mutatis mutandis, (2.2) is satisfied.
Suppose that a projective space P is endowed with parallelisms ℓ and r that are called the left and right parallelism, respectively.We speak of left (right) parallel lines and left (right) parallel classes.According to [17], (P, ℓ , r ) constitutes a projective double space if the following axiom is satisfied.
(DS) For all triangles p 0 , p 1 , p 2 in P there exists a common point of the lines M 1 and M 2 that are defined as follows.M 1 is the line through p 2 that is left parallel to the join of p 0 and p 1 , M 2 is the line through p 1 that is right parallel to the join of p 0 and p 2 .
In case of a projective double space (P, ℓ , r ), each of ℓ and r is referred to as a Clifford parallelism of (P, ℓ , r ).We now generalise this notion.
Definition 3.2.Let (P, ℓ , r ) be a projective double space.A Clifford-like parallelism of (P, ℓ , r ) is a parallelism on P such that By Theorem 3.1, the Clifford-like parallelisms of (P, ℓ , r ) are precisely the blends of ℓ and r .In particular, ℓ and r themselves are the trivial examples of Clifford-like parallelisms of (P, ℓ , r ).
Next, we recall that there exist projective double spaces (P, ℓ , r ) such that ℓ coincides with r .See [8], [14] and [22] for further details, an algebraic characterisation, and geometric properties.Such a double space has only one Clifford-like parallelism, namely ℓ = r .We therefore exclude this kind of double space from our further discussion.
The projective double spaces (P, ℓ , r ) with ℓ r are precisely the ones that can be obtained from quaternion skew fields (see [17], [18], [15], [22]).A detailed account is the topic of the next section.
Finally, we observe that the "left and right Clifford parallelisms" introduced in [3] and defined by an octonion division algebra do not give rise to a projective double space.For further details, see [3] and the references therein.Remark 3.3.In [10, Rem.3.7 and Thm.3.8] the authors gave examples of piecewise Clifford parallelisms with two pieces.Without going into details, let us point out that (in our terminology) these parallelisms arise from a three-dimensional Pappian projective space P that is made into a projective double space in two different ways, say (P, ℓ,1 , r,1 ) and (P, ℓ,2 , r,2 ).Thereby, it has to be assumed that ℓ,1 and ℓ,2 share a single parallel class.The piecewise Clifford parallelisms with two pieces are blends of ℓ,1 and ℓ,2 , but none of these is Clifford-like with respect to any double space structure on P. The proof of the last statement is beyond the scope of this article, since the methods utilised in [10] are totally different from ours.

Clifford-like parallelisms from quaternion skew fields
In this section we deal with a quaternion skew field H with centre F. We thereby stick to the terminology and notation from [4] and [9].Also, we use the abbreviations H * := H \ {0} and F * := F \ {0}.For a detailed account on quaternions we refer, among others, to [28, pp. 46-48] and [29, Ch.I].
The F-vector space H is equipped with a quadratic form H → F, called the norm form, sending q → qq = qq.Here denotes the conjugation, which is an antiautomorphism of the skew field H.The conjugation is of order two and fixes F elementwise.Polarisation of the norm form yields the symmetric bilinear form For any subset X ⊆ H we denote by X ⊥ the set of those quaternions that are orthogonal to all elements of X with respect to • , • .
The projective space P(H) is understood to be the set of all subspaces of the F-vector space H and incidence is symmetrised inclusion.We adopt the usual geometric language: points, lines and planes are the subspaces of vector dimension one, two, and three, respectively.The set of lines of P(H) will be written as L(H).Furthermore, we shall regard ⊥ as a polarity of P(H) sending, for example, any line M to its polar line M ⊥ .For one kind of line this will now be made more explicit.Proof.From L ⊥ = 1 ⊥ ∩ g ⊥ and (4.1), a quaternion u ∈ H belongs to L ⊥ precisely when the following system of equations is satisfied: It is immediate from (4.3) that any u ∈ L ⊥ satisfies (4.2).Conversely, if (4.2) holds for some u ∈ H then g(u + u) = gu + gu = gu + ug ∈ F. Together with g F and u + u ∈ F this forces u + u = 0, whence the system (4.3) is satisfied.
Let M, N ∈ L(H).Then the line M is left parallel to the line N, in symbols M ℓ N, if there is a c ∈ H * with cM = N.Similarly, M is right parallel to N, in symbols M r N, if there is a d ∈ H * with Md = N.The relations ℓ and r make P(H) into a projective double space P(H), ℓ , r , that is, ℓ and r are its Clifford parallelisms (see [15]).In accordance with the terminology and notation from Section 3, each line M ∈ L(H) determines its left parallel class S ℓ (M) and its right parallel class S r (M).All left (right) parallel classes are regular spreads of P (see [4, 4.8 Cor.] or [9,Prop. 4.3]), that is, ℓ and r are regular parallelisms [13,Ch. 26].
For any choice of c, d ∈ H * we can define the F-linear bijection µ c,d : H → H : p → cpd, which acts as a projective collineation on P(H) preserving both the left and the right Clifford parallelism as a straightforward computation shows.Also, µ c,d preserves the norm form of H up to the factor ccdd ∈ F * so that orthogonality of subspaces of H is preserved too.Two particular cases deserve special mention.For d ∈ F * , in particular for d = 1, the mapping µ c,d is a left translation.A right translation arises in a similar way for c ∈ F * .
Let A(H) be the star of lines with centre F1 (with 1 ∈ H), that is, the set of all lines of L(H) passing through the point F1.From an algebraic point of view, each left (right) parallel class has a distinguished representative, namely its only line belonging to A(H).The star A(H) is precisely the set of all two-dimensional F-subalgebras of H or, in other words, the set of all maximal subfields of H.
Let π ℓr denote the equivalence relation on L(H) that is generated by the left and right Clifford parallelism on P(H).If M π ℓr N applies, then we say that M is left-right equivalent to N.
We now present several characterisations of left-right equivalent lines.
Theorem 4.2.Let M 1 , M 2 ∈ L(H) and let L 1 and L 2 be the uniquely determined lines through the point F1 such that L 1 ℓ M 1 and L 2 r M 2 .Then the following are equivalent.
(f) There exists an e ∈ H * with e −1 L 1 e = L 2 .
Proof.(a) ⇒ (b).By the definition of ℓ and r and by virtue of (2.3), we obtain that M 1 π ℓr M 2 implies the existence of an integer n ≥ 1 and elements g 1 , g 2 , . . ., g 2n such that . By our assumption, there exists a line M, say, belonging to S ℓ (M 1 )∩ S r (M 2 ).Also, there is a left translation µ c 1 ,1 taking M to L 1 , i.e., c 1 M = L 1 .Since µ c 1 ,1 preserves not only the left and right Clifford parallelism but also the orthogonality of lines in both directions, it suffices to verify that To this end we pick a quaternion g ∈ L 1 \ F, which is maintained throughout this part of the proof.First, we take any line N ∈ S ℓ (L 1 ) ∩ S r (L 1 ).For all u ∈ N * we obtain from 1 ∈ L 1 that N = uL 1 = L 1 u.Thus the inner automorphism µ u −1 ,u of H restricts to an automorphism of L 1 .There are two possibilities.
Remark 4.9.The group Γ of all collineations of P(H) that preserve both the left and the right parallelism was described in [25,Thm. 1] in terms of the factor group H * /F * , which thereby serves as a model for the point set P(H) by identifying F * c with Fc for all c ∈ H * .By [26,Prop. 4.1 and Prop.4.2], a collineation γ of P(H) belongs to Γ if, and only if, γ can be induced by an F-semilinear transformation of H that is the product of an automorphism of the skew field H and a map µ c,d for some c, d ∈ H * (see also [3,Thm. 4.3]).
In particular, the maps µ c,d induce exactly the F-linear part of the group Γ.If Char F 2 then we know by [19, (4.16)] that they induce precisely the proper motions of the elliptic 3-space P(H), so the classes of left-right equivalent lines turn out to be the line orbits under the action of the elliptic proper motion group.
The following result describes all Clifford-like parallelisms of P(H), ℓ , r .Theorem 4.10.Let A(H) be the subset of all lines of P(H) through the point F1.
So, substituting in (4.4) gives Now, by comparing (4.5) with (4.6), we obtain The given parallelism is a blend of ℓ and r by Theorem 3.1.Thus allows a construction as described in Proposition 2.4 (a) using ℓ , r , and some subset, say D, of L(H).Replacing D with the set does not alter this result, as has been pointed out in Proposition 2.4 (c).The first part of the current proof shows that we also get the parallelism by applying the construction from (a) to the set D from (4.8).
(c) By the first part of the proof, we obtain F and F from D and D, respectively, also via the construction in Proposition 2.4 (a).In our current setting the condition (2.6) simplifies to since L(H) ℓr = ∅ by Corollary 4.3.From (4.7), equation (4.9) is equivalent to F = F.It therefore suffices to make use of Proposition 2.4 (c), with (2.6) to be replaced by F = F, in order to complete the proof.
Remark 4.11.Theorem 4.10 (a) was sketched without a strict proof in [4, 4.13].However, there are some formal differences to our approach, as we avoid the indicator lines of regular spreads that have been used there.Our set of lines A(H) is, from an algebraic point of view, the family of all quadratic extensions L of F with F ⊆ L ⊆ H from [4, 4.13].In this way, our F turns into a family of subfields of H. Equation (4.4) guarantees that no subfield in F is F-isomorphic to a subfield in A(H) \ F. The latter condition is mentioned in the sketch of proof from [4, 4.13], but is missing there at that point, where the family F is fixed for the first time.
(F-isomorphic subfields of H are termed as being "conjugate" in [4].) Below we shall make use of the ordinary quaternion algebra over a formally real field F, i.e. −1 is not a square in F. This kind of algebra will be denoted as (K/F, −1).According to [28, pp. 46-48] it arises (up to F-isomorphism) in the following way.The field F is extended to K := F(i), where i is a square root of −1 ∈ F. One defines (K/F, −1) as the subring of the ring of 2 × 2 matrices over K consisting of all matrices and identifies any x ∈ F with the matrix diag(x, x) ∈ (K/F, −1).The F-algebra (K/F, −1) is a skew field if, and only if, −1 is not a sum of two squares in F. If the latter condition applies then (K/F, −1) is called the ordinary quaternion skew field over F. For example, an ordinary quaternion skew field exists over any formally real Pythagorean field.We recall that a field is Pythagorean when the sum of any two squares is a square as well (see e.g.[20, p. 204]).
Theorem 4.12.Let H be a quaternion skew field with centre F. Then the following are equivalent.The set A(H) \ F comprises precisely the inseparable quadratic extensions of F that are contained in H.We get F = D, since the group of inner automorphisms of H, in its natural action on A(H), leaves both D and A(H) \ D invariant.Both F and A(H) \ F are non-empty; see, among others, [6, pp. 103-104] or [28, pp. 46-48].So D gives rise to a Clifford-like parallelism of P(H), ℓ , r other than ℓ and r .mutually opposite reguli.The third equation in (4.10) gives R ′ (L 3 , L 1 ) ⊆ S(L 3 ).Taking into account that P(H), , ′ admits a description in terms of some quaternion skew field, we apply Theorem 4.2 and get |S(L 3 ) ∩ S ′ (L 3 )| ≤ 2, whereas R ′ (L 3 , L 1 ) ⊆ S(L 3 ) ∩ S ′ (L 3 ) has |F| + 1 elements, a contradiction.As a consequence of Theorems 4.12 and 4.15, we obtain the following: Corollary 4.16.A projective double space P(H), ℓ , r , where H does not satisfy condition (a) from Theorem 4.12, admits Clifford-like parallelisms that are not Clifford w.r.t.any double space structure on P.

Lemma 2 . 1 .
Let π be an equivalence relation on L and B ⊆ L. Then the following are equivalent.(a) B admits a partition by π-classes.(b) {C(X) | X ∈ B} is the only partition of B by π-classes.(c) B = X∈B C(X).(d)L \ B admits a partition by π-classes.Proof.(a) ⇒ (b).Let Σ be a partition of B by π-classes.Then Σ coincides with the partition given in (b).(b) ⇒ (c).This is obvious.(c) ⇒ (d).It suffices to observe that {C(X) | X ∈ L \ B} is a partition of L \ B by π-classes.(d) ⇒ (a).The existence of a partition, say Σ ′ , of L \ B by π-classes implies that Π \ Σ ′ is a partition of B by π-classes.

Lemma 2 . 3 .Proposition 2 . 4 .
Let π 1 and π 2 be equivalence relations on L and denote by π 12 the equivalence relation generated by π 1 and π 2 .Furthermore, let B ⊆ L. Then the following statements are equivalent.(a) B admits a partition by π 12 -classes.(b) B admits a partition by π 1 -classes and a partition by π 2 -classes.Proof.Let (a) be satisfied.For each X ∈ B, we have C 1 (X)∪C 2 (X) ⊆ C 12 (X) ⊆ B, where the second inclusion follows by applying Lemma 2.1 to an existing partition of B by π 12 -classes.This forces B ⊆ Y∈B C 1 (Y) ⊆ B and B ⊆ Z∈B C 2 (Z) ⊆ B. These two formulas in combination with Lemma 2.1 establish (b).Conversely, let us choose some M ∈ B. Then, for all N ∈ C 12 (M), there is a finite sequence as in (2.3), whence N ∈ B. Thus C 12 (M) ⊆ B. This shows B ⊆ X∈B C 12 (X) ⊆ B, and Lemma 2.1 provides the existence of a partition of B by π 12 -classes.Let π 1 and π 2 be equivalence relations on L. Furthermore, denote by π 12 the equivalence relation generated by π 1 and π 2 .(a)Upon choosing any subset D of L we let B := X∈D C 12 (X).

Lemma 4 . 1 .
For any line L = F1 ⊕ Fg, where 1 ∈ H and g ∈ H \ F, the line L ⊥ is the set of all u ∈ H subject to the condition ug = gu.(4.2)

( a )
Upon choosing any subset D of A(H) we let F := X∈D, c∈H * c −1 Xc.(4.4)ThenΠ F := S ℓ (L) | L ∈ F ∪ S r (L) | L ∈ A(H) \ F (4.5)is the set of parallel classes of a Clifford-like parallelism F , say, of the projective double space P(H), ℓ , r .(b)Conversely, any Clifford-like parallelism of P(H), ℓ , r arises according to (a) from at least one subset of A(H).(c) Let D and D be subsets of A(H).Applying the construction from (a) to D and D gives parallelisms F and F, respectively.Then F coincides with F if, and only if, F = F. Proof.(a) We apply the construction from Proposition 2.4 (a) to D; thereby we replace π 1 and π 2 with ℓ and r , respectively.So, starting with B := X∈D C ℓr (X), we finally arrive at the partition Π B from (2.5), whose associated equivalence relation on L(H) is a blend of ℓ and r .By Theorem 3.1, this Π B is the set of parallel classes of a Clifford-like parallelism of P(H), ℓ , r .Each of its parallel classes has a unique line in common with A(H).Therefore Π B = S ℓ (L) | L ∈ B ∩ A(H) ∪ S r (L) | L ∈ A(H) \ B .(4.6) Theorem 4.2 and Corollary 4.7 show that (a) F is a formally real Pythagorean field, and H is the ordinary quaternion skew field over F. (b) All maximal subfields of H are mutually F-isomorphic.(c) The Clifford parallelisms ℓ and r are the only Clifford-like parallelisms of the projective double space P(H), ℓ , r .Proof.(a) ⇔ (b).This was established in [7, Thm. 1 and Lemma 1] (but note that the definition of Pythagorean field used there is slightly different from ours).See also [3, Thm.9.1] for a proof in a more general situation.(b) ⇔ (c).From Theorem 4.2, all maximal subfields of H are F-isomorphic if, and only if, for all L ∈ A(H) we have A(H) = c∈H * c −1 Lc.The last equation holds precisely when there are only two possibilities for the set F appearing in (4.4), namely either F = A(H) or F = ∅.This in turn is equivalent, by Theorem 4.10, to saying that ℓ and r are the only Clifford-like parallelisms of P(H), ℓ , r .We continue by giving some explicit examples of Clifford-like parallelisms using the construction from Theorem 4.10 (a).Example 4.13.Let Char H = 2.We define D := {L ∈ A(H) | L is a separable extension of F}.