# Cayley algebras give rise to $q$-Fano planes over certain infinite fields
and $q$-covering designs over others

Vincent van der Noort

###### Abstract.

Let $F$ be a field. A $2$-$(7,3,1)_{F}$-subspace design or _$q$ -Fano plane_
over $F$, is a $7$-dimensional vector space $V$ over $F$ together with a
collection $\mathfrak{B}$ of $3$-dimensional subspaces of $V$ such that every
two-dimensional subspace of $V$ is contained in exactly one element $B$ of
$\mathfrak{B}$. The question of existence of such a subspace design over any
field has been open since the 1970s and has attracted considerable attention
in the special case that $F$ is finite. Here we show the existence of
$2$-$(7,3,1)_{F}$-subspace designs over a collection of _infinite_ fields $F$,
including (among others) the fields $\mathbb{Q}$ and $\mathbb{R}$ and the
function fields $\mathbb{F}_{q}(x,y,z)$ with $q$ odd.

The space $V$ is taken to be the 7-dimensional space of imaginary elements in
a non-associative, 8-dimensional Cayley division algebra $O$ over $F$ and the
collection $\mathfrak{B}$ consists of the intersections with $V$ of all
4-dimensional (quaternion) subalgebras of $O$. We will present all relevant
facts about quaternion and Cayley algebras in a nearly self-contained fashion.

The second part of the paper studies what happens if we apply the same
procedure to the split Cayley algebra over $F$ rather than a Cayley division
algebra. (This is relevant since over a large collection of fields, including
$\mathbb{C}$, $\mathbb{F}_{p^{n}}$ and $\mathbb{Q}_{p}$ for all $p$, only
split Cayley algebras exist.) By identifying all four-dimensional subalgebras
of these algebras, we show that in this case our construction still yields an
inclusion minimal $(7,3,2)$ $q$-covering design. That is: every two-
dimensional subspace of $V$ is contained in at least one element of the
resulting set $\mathfrak{B}$ of three dimensional subspaces of $V$ and no
proper subset of $\mathfrak{B}$ has this property. However none of these
$q$-covering designs are $q$-Fano planes. In the case that $F$ is finite we
compute the number of elements of $\mathfrak{B}$.

For general $F$ of characteristic $\neq 2$, we also give a purely
combinatorial ‘direct’ construction (not mentioning the underlying algebra
structure) of our $q$-Fano planes and $q$-covering designs for an abstract
7-dimensional $F$-vector space $V$ by identifying the collection
$\mathfrak{B}$ as a subvariety of the Grassmanian $\operatorname{Gr}_{3}(V)$
defined entirely in terms of the classical Fano plane.

###### Contents

  1. 1 Introduction
    1. 1.1 Subspace designs, $q$-covering designs and $q$-Fano planes
    2. 1.2 Statement of the main result and some limitations
    3. 1.3 Real numbers, complex numbers, quaternions, octonions
      1. 1.3.1 Complex numbers and quaternions
      2. 1.3.2 Octonions and the classical Fano plane
    4. 1.4 Overview of the article
    5. 1.5 Acknowledgements
  2. 2 Preliminaries from algebra
    1. 2.1 Conventions and definitions
    2. 2.2 A theorem of Artin
    3. 2.3 A particularly nice class of algebras
    4. 2.4 The Dickson Double
    5. 2.5 The Cayley-Dickson algebras
    6. 2.6 Multiplication table and relation to the Fano plane
    7. 2.7 Which fields allow Cayley division algebras?
    8. 2.8 Some structure theory
  3. 3 $q$-Fano planes from Cayley Division algebras
  4. 4 The subalgebras generated by two-dimensional subspaces of $\operatorname{Im}(O)$
    1. 4.1 Subspaces generating three-dimensional subalgebras
    2. 4.2 Multiplication table of the split Cayley-Algebra
    3. 4.3 A classification of two-dimensional subspaces of $\operatorname{Im}(O)$
    4. 4.4 The algebras generated by subspaces of type D
    5. 4.5 The algebras generated by subspaces of type M
    6. 4.6 The algebras generated by subspaces of type J
    7. 4.7 No $q$-Fano planes from split Cayley algebras
  5. 5 $q$-Covering designs from split Cayley Algebras
  6. 6 Relation to associativity
  7. 7 Geometric reformulation of the main results
  8. 8 Quantizing the Fano plane: combinatorial reformulation of the main results
  9. 9 Quantifying the results of section 4 in the finite field case
    1. 9.1 Notational conventions
    2. 9.2 Summary of the results
    3. 9.3 The subalgebras of type M4
    4. 9.4 Completing Table 7
    5. 9.5 Subalgebras containing subspaces of type Z
    6. 9.6 Final computations
    7. 9.7 The mysterious equality (68)

##  1\. Introduction

###  1.1. Subspace designs, $q$-covering designs and $q$-Fano planes

A $t$-$(v,k,\lambda)$ _subspace design_ over a field $F$ is a collection
$\mathcal{B}$ of $k$-dimensional linear subspaces (called _blocks_) of a
$v$-dimensional $F$-vector space $V$, with the property that every
$t$-dimensional linear subspace of $V$ is contained in exactly $\lambda$
elements of the set $\mathcal{B}$. They are considered the $q$-analogue of the
more familiar (combinatorial) _designs_ which are defined in a similar way but
with sets replacing $F$-vector spaces. Subspace designs over $F$ with
$\lambda=1$ are called _$q$ -Steiner systems_ over $F$, by analogy with the
ordinary Steiner systems.

According to Kiermaier and Laue [KL15], subspace designs were first introduced
in 1973 by Cameron [Cam74] but the first explicit construction of non-trivial
subspace designs appeared only in 1989 in the work of Thomas [Tho87].
Meanwhile the existence of $q$-Steiner systems remained open until the
construction, in 2013, of various $2$-$(13,3,1)$ subspace designs over
$\mathbb{F}_{2}$ by Braun, Etzion, Ostergard, Vardy and Wasserman [BEO+16].

The existence of a $2$-$(7,3,1)$ subspace design, also known as a _$q$ -Fano
plane_ over any finite field is unknown. Severe restrictions on a hypothetical
$q$-Fano plane over the field $\mathbb{F}_{2}$ have been proposed by various
authors. In particular, from results of Braun, Kiermaier, Kurz, Nakic and
Wasserman [BKN15] [KKW18] we know that the order of the automorphism group of
such a design would be at most 2.

Kiermaier [Kie16] has called the existence of a $q$-Fano plane over any finite
field the ‘most important open problem in $q$-analogs of designs’.

In this paper we will show the existence of $q$-Fano planes over certain
_infinite_ fields, most notably all subfields of the field $\mathbb{R}$ of
real numbers and the function fields $K(\alpha,\beta,\gamma)$ where $K$ is any
field of characteristic unequal to 2.

A concept closely related to that of subspace designs is that of a
$q$-covering design. A $q$-covering design with parameters $(v,k,t)$ on a
$v$-dimensional vector space $V$ over a field $F$ is a set $\mathfrak{B}$ of
$k$-dimensional subspaces of $V$ such that every $t$-dimensional subspace of
$V$ is contained in _at least_ one elements of $\mathfrak{B}$. Since the set
$\operatorname{Gr}_{k}(V)$ of all $k$-dimensional subspaces of $V$ is a rather
trivial example of a $q$-covering design it stands to reason to consider the
$q$-covering designs more interesting when they have a smaller number of
blocks. By this measure $t$-$(v,k,1)$-subspace designs are the most
interesting $(v,k,t)$-$q$-covering designs. It is easy to see that over a
finite field no $q$-covering design can have a smaller number of blocks than a
hypothetical $q$-Steiner system over the same field with the same parameters
would have: this number can be computed as the number of $t$-dimensional
subspaces of $V$ divided by the number of $t$-dimensional subspaces in a given
block and neither number depends on the chosen set of blocks. However since
for most parameters the existence of subspace designs over finite fields is
unknown, it is an open question what is the minimum number $\mathcal{C}_{q}$
of blocks in a $q$-covering design over a field with $q$ elements.
$q$-Covering designs have application in network design as described by
Lambert [Lam13], who also gives upper and lower bounds on the numbers
$\mathcal{C}_{q}$.

###  1.2. Statement of the main result and some limitations

In the current paper we will show the existence of $q$-Fano planes (that is
$2$-$(7,3,1)$-subspace designs) over certain infinite fields, including the
function fields $K(\alpha,\beta,\gamma)$ where $K$ is any field of
characteristic unequal to 2, and all subfields of the field $\mathbb{R}$ of
real numbers, so in particular $\mathbb{R}$ itself. In this introduction we
focus on the latter case as it is the easiest to work with.

Formulated in purely combinatorial terms we prove:

######  Theorem 1.1.

Let $F$ be field of characteristic $\neq 2$ and let
$(\mathcal{V},\mathcal{L})$ be a Fano-plane with vertex set $\mathcal{V}$ and
line set $\mathcal{L}\subset\mathcal{V}^{3}$. We fix an automorphism $\phi$ of
$(\mathcal{V},\mathcal{L})$ of order 7 and choose a cyclic ordering on the
three elements of $l$ for each $l\in\mathcal{L}$. The orderings must be
‘compatible’ in the sense that application of $\phi$ will preserve the
orderings on the lines. A rather standard way to achieve this is to label the
elements of $\mathcal{V}$ $v_{0},\ldots,v_{6}$, and define the ordered lines
to be the (cyclically ordered) triples $(v_{n},v_{n+1},v_{n+3})$ where the
indices are read modulo 7. (Here the automorphism $\phi$ amounts to cyclically
permuting the indices.)

We define $V$ to be the seven-dimensional $F$-vector space with basis
$\mathcal{V}$. Let $W=\bigwedge^{3}V$, so $W$ is thirtyfive-dimensional. Let
$\Delta=v_{0}\wedge\phi(v_{0})\wedge\phi^{2}(v_{0})\ldots\wedge\phi^{6}(v_{0})\in\bigwedge^{7}V\cong
F$ where $v_{0}$ is some element of $\mathcal{V}$. Note that, since 7 is an
odd number, $\Delta$ does not depend on the choice of $v_{0}\in\mathcal{V}$.
Similarly, for each $l\in\mathcal{L}$ let $w_{l}\in W$ be the wedge-product of
the three points in $l$ in the given cyclic order. (Again this is well-defined
due to the oddness of 3.) These data in turn define, for every
$v\in\mathcal{V}$ a linear functional $\eta_{v}\colon W\to F$ by
$v\wedge(\sum_{l\in\mathcal{L}}w_{l})\wedge w=\eta_{v}(w)\Delta$.

Let $\operatorname{Gr}_{3}(V)$ be the set of three-dimensional subspaces of
$V$ and $\operatorname{Gr}_{1}(W)$ the set of one-dimensional subspaces of
$W$. Let $\psi\colon\operatorname{Gr}_{3}(V)\to\operatorname{Gr}_{1}(W)$ be
the Plücker embedding. Recall that $\psi$ sends a three-dimensional subsapce
$B\subset V$ to the line $F(b_{1}\wedge b_{2}\wedge b_{3})\in W$ where
$\\{b_{1},b_{2},b_{3}\\}$ is _any_ basis of $B$; this assignment is well-known
to be both well defined and injective.

Then the set
$\mathfrak{B}=\\{B\in\operatorname{Gr}_{3}(V)\colon\psi(B)\subset\ker\eta_{v}\textnormal{
for all }v\in\mathcal{V}\\}$ is an inclusion-minimal $q$-covering design with
parameters $(7,3,2)$ on $V$. In other words: every two-dimensional subspace of
$V$ is contained in at least one element of $\mathfrak{B}$ and no proper
subcollection of $\mathfrak{B}$ has this property.

Moreover, if $F$ is a subfield of the field of real numbers $\mathbb{R}$ (such
as $\mathbb{R}$ itself) then $\mathfrak{B}$ is a $q$-Fano plane, that is: if
$F\subset\mathbb{R}$ we have that every two-dimensional subspace of $V$ is
contained in _exactly one_ element of $\mathfrak{B}$.

Further, if $F$ is a finite field with $q$ elements (recall that we required
$\operatorname{char}(F)$ and hence $q$ to be odd) then the $q$-covering design
has $\genfrac{[}{]}{0.0pt}{}{6}{2}_{q}$ blocks.

Thus, in the finite field case, the number of blocks in the subspace design
has the same leading term ($q^{8}$) as the number
$\genfrac{[}{]}{0.0pt}{}{7}{2}_{q}/\genfrac{[}{]}{0.0pt}{}{3}{2}_{q}$ of
blocks in a hypothetical $q$-Fano plane over $F$, but exceeds this number by
$q[7]_{q}+q^{4}$.

We stress that the failure of the above construction to produce a $q$-Fano
plane over finite fields is really due to the finiteness of the field and not
to it having positive characteristic. As remarked above a different but
closely related construction yields a $q$-Fano plane over fields
$F=K(\alpha,\beta,\gamma)$ which, in case $K$ has a finite (but odd) number of
elements, is an infinite field of positive characteristic. A more general
version of Theorem 1.1, which encompasses both cases as well as a number of
others is given in Section 8 (Thm. 8.1).

Besides finite fields there is a another well known class of fields which are
necessarily infinite but over which our construction fails to produce a
$q$-Fano plane: fields that are algebraically closed. A precise criterion for
over which fields our construction gives a $q$-Fano plane is given in Section
2.7.

Working over infinite fields robs us of the opportunity of comparing which of
two $q$-covering designs is smaller by counting the number of blocks. It still
seems fair however to say that a $q$-covering design $\mathfrak{B}^{\prime}$
is smaller than a $q$-covering design $\mathfrak{B}$ with the same parameters
when $\mathfrak{B}^{\prime}\subset\mathfrak{B}$. In that sense it is
encouraging that for the $q$-covering designs given by Theorem 1.1 no such
‘subdesigns’ $\mathfrak{B}^{\prime}$ exist. At the same time however, our
construction shows that this ‘inclusion-minimality’ is of limited use as a
means to compare $q$-covering designs: the more general version of Thm 1.1
(Thm 8.1) implies that inclusion-minimal $(7,3,2)$-$q$-covering designs that
are _not_ $q$-Fano planes can be constructed over _any_ field $F$, including
fields $F$ where $q$-Fano planes do exist, such as $\mathbb{R}$. Also, in the
case of $F=\mathbb{F}_{q}$ the fact that the $q$-covering design of Thm 1.1 is
inclusion minimal does not by itself prove that the upper bound of
$\genfrac{[}{]}{0.0pt}{}{6}{2}_{q}=q^{8}+q^{7}+2q^{6}+2q^{5}+3q^{4}+2q^{3}+2q^{2}+q+1$
on $\mathcal{C}_{q}(7,3,2)$ is sharp. In fact it is not: carrying out the
construction in Section 4.4.2 of Lambert’s thesis [Lam13] for parameters
$(7,3,2)$ one arrives at a lower upper bound of
$q^{8}+2q^{6}+3q^{4}+q^{3}+2q^{2}+q+1$.

In the present work, we will not discuss fields of characteristic 2 nor
$q$-covering designs with different parameters than $(7,3,2)$.

###  1.3. Real numbers, complex numbers, quaternions, octonions

Theorem 1.1 was formulated in combinatorial terms, starting with the classical
Fano plane and ‘quantizing’ it into the $q$-Fano plane. This was done partly
order to obtain a self-contained statement and partly in the hope that it
would allow my future self or the reader to guess the right generalization to
designs with other parameters than $2$-$(7,3,1)$. However most of the proof
(and of the current paper) does not refer to the classical Fano plane at all
and instead revolves around _Cayley algebras_ , the generalization to
arbitrary fields of an extension of the real numbers called _octonions_ ,
named so since they occupy an 8-dimensional real vector space. It is with
pleasure that I quote Baez [Bae02]:

> ‘There are exactly four normed division algebras: the real numbers
> ($\mathbb{R}$), complex numbers ($\mathbb{C}$), quaternions ($\mathbb{H}$),
> and octonions ($\mathbb{O}$). The real numbers are the dependable
> breadwinner of the family, the complete ordered field we all rely on. The
> complex numbers are a slightly flashier but still respectable younger
> brother: not ordered, but algebraically complete. The quaternions, being
> noncommutative, are the eccentric cousin who is shunned at important family
> gatherings. But the octonions are the crazy old uncle nobody lets out of the
> attic: they are nonassociative.’

It is because of this non-associativity (and subsequent obscurity) that we
spend a sizable portion of the paper on presenting the preliminaries. Section
2 will recall in detail the theory of quaternion and Cayley algebras over
general fields of characteristic $\neq 2$. However in this introduction we
will first discuss in a more concrete fashion the real algebras $\mathbb{H}$
and $\mathbb{O}$ that stand at the cradle of this theory. We’ll spend more
time at the quaternions as they are easier to work with and already illustrate
the most important concepts.

####  1.3.1. Complex numbers and quaternions

The octonions $\mathbb{O}$ were independently discovered by Graves in 1843 and
Cayley in 1845, both building on Hamilton’s 1843 discovery of the quaternions
$\mathbb{H}$. ([Bae02]). Hamilton was motivated by geometry ([Bae02]):
understanding how rotations and translations in two-dimensional geometry could
be understood as multiplication and addition of complex numbers, Hamilton
sought an algebraic structure that would give a similar description for
rotations and translations in three dimensions. The quaternions achieve this
exceptionally well ([Bae02], [CS03]) and their construction from the reals is
very similar to that of the complex numbers. Indeed, where the complex numbers
are obtained by attaching a square root $i$ of $-1$ to $\mathbb{R}$, the
quaternions are obtained by attaching _three_ square roots of $-1$, called
$i,j,k$ and which are related by

$ij=k;\qquad ji=-k$

and cyclic permutions of those equations, so

$jk=i;\qquad kj=-i;\qquad ki=j;\qquad ik=-j.$

A _quaternion_ is any element of the four-dimensional vectorspace spanned by
$1,i,j,k$ with multiplication implied by the above equations and
$i^{2}=j^{2}=k^{2}=-1$. Before moving on to the octonions we discuss a few
properties of the quaternions which will be important in the sequel.

First, as mentioned before, the quaternions form a _division algebra_ , which
for the context of this paper can be understood to mean that there are no
zero-divisors:

(1) $ab=0\Rightarrow a=0\textrm{ or }b=0.$

(The precise relation between this property and what one would ordinarily call
division is discussed in Section 2.1.) A standard way to see that this
property holds in the complex numbers is to show that the Euclidian norm
$\|.\|$ on $\mathbb{C}\cong\mathbb{R}^{2}$ is multiplicative: for complex
numbers $x,y$ we have that $\|xy\|=\|x\|\|y\|$, assigning to any hypothetical
counterexample to (1) in $\mathbb{C}$ a corresponding example in $\mathbb{R}$
where we already know that (1) holds. Multiplicativity of the norm in turn is
achieved by expressing the square of the norm as the product of a complex
number $a=\alpha+\beta i$ by its complex conjugate $a^{*}\coloneqq\alpha-\beta
i$ and noticing that taking complex conjugates is an automorphism of
$\mathbb{C}$: $(xy)^{*}=x^{*}y^{*}=y^{*}x^{*}$ so that

(2)
$\|xy\|^{2}=xyy^{*}x^{*}=\|y\|^{2}xx^{*}=\|y\|^{2}\|x\|^{2}=\|x\|^{2}\|y\|^{2}$

Interestingly the proof in the quaternion case is almost exactly the same.
Here we define the analogue of complex conjugation $x\mapsto x^{*}$ by

(3) $(\alpha+\beta i+\gamma j+\delta k)^{*}=\alpha-\beta i-\gamma j-\delta k$

(which reduces to the ordinary complex conjugation when $\delta=\gamma=0$) and
we notice that again $xx^{*}=\|x\|^{2}$, where $\|.\|$ denotes the standard
Euclidean norm on $\mathbb{R}^{4}\cong\mathbb{H}$. Here the non-commutativity
in the definition of the multiplication nicely helps us to get rid of the
cross-terms and end up with a sum of squares. At the same time this non-
commutativity prevents the conjugation map from being an automomorphism this
time. However, it does make it into an _anti-automorphsim_ , that is: we have

$(xy)^{*}=y^{*}x^{*}$

and hence (2) goes through unmodified and we conclude that indeed $\mathbb{H}$
is a division algebra.

The second feature of $\mathbb{H}$ we want to stress, before moving on to
discussing octonions, is its richness in subalgebras isomorphic to
$\mathbb{C}$. We ‘created’ the quaternions by attaching three squareroots
($i,j,k$) of $-1$ to $\mathbb{R}$. But in doing so we introduced infinitely
many more. Every element $i^{\prime}$ of Euclidean norm 1 in the three-
dimensional space $\operatorname{Im}(\mathbb{H})=\operatorname{span}(i,j,k)$
is a squareroot of $-1$. (This follows directly from the claim above that
$xx^{*}=\|x\|^{2}$: for every $x\in\operatorname{Im}(\mathbb{H})$ we have by
(3) that $x^{*}=-x$ and hence $x^{2}=-\|x\|^{2}\in\mathbb{R}_{\leq 0}$.)
Elements of $\operatorname{Im}(\mathbb{H})$ are called _imaginary
quaternions_.

For every imaginary norm-one quaternion $i^{\prime}$, the two-dimensional
subspace
$\mathbb{C}^{\prime}\coloneqq\operatorname{span}(1,i^{\prime})\subset\mathbb{H}$
is isomorphic (not just as a field, but also as an $\mathbb{R}$-algebra) to
$\mathbb{C}$; where complex conjugation is just the restriction to
$\mathbb{C}^{\prime}$ of the quaternion conjugation (3) and where the one-
dimensional space $\operatorname{Im}(\mathbb{C}^{\prime})$ (the scalar
multiples of $i^{\prime}$) is just the intersection of $\mathbb{C}^{\prime}$
with $\operatorname{Im}(\mathbb{H})$.

More generally, the vector space decomposition
$\mathbb{H}=\mathbb{R}1\oplus\operatorname{Im}(\mathbb{H})$ is very similar to
the decompostion $\mathbb{C}=\mathbb{R}1\oplus\operatorname{Im}(\mathbb{C})$:
in both cases the first summand consists of all elements whose square is a
positive real number and the second of all elements whose square is a negative
real number and in both cases the linear map ∗ which acts as $1$ on the first
summand and as $-1$ on the second is an anti-automorphism that reproduces the
euclidean norm through $x^{*}x=\|x\|^{2}$. Of course, setting
$\operatorname{Im}(\mathbb{\mathbb{R}})=\\{0\\}$, the same facts hold in
$\mathbb{R}$ as well.

####  1.3.2. Octonions and the classical Fano plane

Given the above the following will not come as a surprise. The octonions
$\mathbb{O}$ are formed by attaching to $\mathbb{R}$ no less than $7$
squareroots of $-1$ (following Conway-Smith [CS03] we will call them
$e_{0},\ldots,e_{6}$) that, for $i\neq j$ satisfy
$e_{i}e_{j}=-e_{j}e_{i}\in\operatorname{Im}(\mathbb{O})$ where
$\operatorname{Im}(\mathbb{O})=\operatorname{span}(e_{0},\ldots,e_{6})$. From
here it follows as before that
$\mathbb{O}=\mathbb{R}1\oplus\operatorname{Im}(\mathbb{O})$ and that by
defining ∗ as the linear involution which acts as the identity on the first
summand and as minus the identity on the second we have that ∗ is an anti-
automorphism satisfying $x^{*}x=\|x\|^{2}$ where $\|.\|$ denotes the Euclidean
norm on $\mathbb{R}^{8}$. What all this does not tell us is how exactly the
$e_{i}$ interact. The multiplication table is given in Table 1, but the
structure becomes a bit clearer in the following description taken from
[CS03]:

$e_{i}e_{i+1}=e_{i+3}\qquad e_{i+1}e_{i}=-e_{i+3}$

Where indices are read $\mod 7$, together with cyclic permutations of these
equations, so

$e_{i+1}e_{i+3}=e_{i};\qquad e_{i+3}e_{i+1}=-e_{i};\qquad
e_{i+3}e_{i}=e_{i+1};\qquad e_{i}e_{i+3}=-e_{i+1};.$

It is well known that the cyclic group on seven elements together with the set
of all seven triples $(i,i+1,i+3)$ from that group forms a model of the
classical Fano plane. Indeed the Steiner triple system property of the Fano
plane is why the above equations (together with $e_{i}^{2}=-1$) are enough to
determine the multiplication on any pair of generators (and hence on all of
$\mathbb{O}$). The Fano plane structure also sheds some light on the infamous
non-associativity of $\mathbb{O}$. For every ‘line’ $(e_{i},e_{i+1},e_{i+3})$
we have that products of elements in their span _is_ associative: in fact we
see from the above that $\operatorname{span}(1,e_{i},e_{i+1},e_{i+3})$ is a
subalgebra of $\mathbb{O}$ isomorphic to $\mathbb{H}$. On the other hand, for
triples of basis elements _not_ on a Fano-line, associativity fails (mildly):
for instance for the triple $(e_{0},e_{1},e_{2})$ we have that
$(e_{0}e_{1})e_{2}=e_{3}e_{2}=-e_{5}$ while
$e_{0}(e_{1}e_{2})=e_{0}e_{4}=e_{5}$.

While the seven Fano-lines each determine a quaternion subalgebra of
$\mathbb{O}$, the total number of such subalgebras is much larger. Analogous
to how each line in $\operatorname{Im}(\mathbb{H})$ generates a subalgebra of
$\mathbb{H}$ isomorphic to $\mathbb{C}$ we have in $\mathbb{O}$ the following
result:

######  Proposition 1.2.

Each two-dimensional subspace of $\operatorname{Im}(\mathbb{O})$ generates a
four-dimensional subalgebra of $\mathbb{O}$ isomorphic to $\mathbb{H}$.

The proof of this proposition is non-trivial, but we won’t discuss it here as
we will prove a more general result down the line.

We _will_ discuss here however a number of interesting consequences of
Proposition 1.2. First we notice:

######  Corollary 1.3.

Every subalgebra of $\mathbb{O}$ generated by 2 elements is associative.

Algebras with this property are called _alternative_. Since it is not hard to
see that the subalgebras generated by 2 elements are closed under the
involution ∗ as well, we conclude from Corollary 1.3 (and the already
estabilished properties of ∗) that (2) holds for all $x,y$ in spite of
$\mathbb{O}$ not being associative. Hence we conclude:

######  Corollary 1.4.

The multiplication of $\mathbb{O}$ satisfies $\|xy\|=\|x\|\|y\|$ where $\|.\|$
is the Euclidean norm on $\mathbb{R}^{8}$ and hence $\mathbb{O}$ is a division
algebra.

Of more interest to us, however, is the following corollary to Proposition
1.2:

######  Theorem 1.5.

Let $\mathcal{H}$ be the collection of all four-dimensional subalgebras of
$\mathbb{O}$ that are isomorphic to $\mathbb{H}$ and let
$\mathcal{B}=\\{\operatorname{Im}(H)\colon H\in\mathcal{H}\\}$ be the
collection of their three-dimensional subspaces of imaginary elements. Then
each $B\in\mathcal{B}$ is a three-dimensional subspace of the seven-
dimensional real vector space $\operatorname{Im}(\mathbb{O})$ and the pair
$(\operatorname{Im}(\mathbb{O}),\mathcal{B})$ is a $q$-Fano plane over
$\mathbb{R}$.

Doubtless, Proposition 1.2 has been known to experts in the field for a long
time, but to the best of my knowledge nobody has openly made the connection
with $q$-Fano planes before. The reason that the current article is not just
two lines long, is that we want to understand what happens over fields
different from $\mathbb{R}$.

Table 1. Multiplication table of $\mathbb{O}$ with respect to the basis $e_{0},\ldots,e_{6}$ described in Section 1.3.2 | $1$ | $e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$
---|---|---|---|---|---|---|---|---
$1$ | $1$ | $e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$
$e_{0}$ | $e_{0}$ | $-1$ | $e_{3}$ | $e_{6}$ | $-e_{1}$ | $e_{5}$ | $-e_{4}$ | $-e_{2}$
$e_{1}$ | $e_{1}$ | $-e_{3}$ | $-1$ | $e_{4}$ | $e_{0}$ | $e_{2}$ | $e_{6}$ | $-e_{5}$
$e_{2}$ | $e_{2}$ | $-e_{6}$ | $-e_{4}$ | $-1$ | $e_{5}$ | $e_{1}$ | $-e_{3}$ | $e_{0}$
$e_{3}$ | $e_{3}$ | $e_{1}$ | $-e_{0}$ | $-e_{5}$ | $-1$ | $e_{6}$ | $e_{2}$ | $-e_{4}$
$e_{4}$ | $e_{4}$ | $-e_{5}$ | $e_{2}$ | $-e_{1}$ | $-e_{6}$ | $-1$ | $e_{0}$ | $e_{3}$
$e_{5}$ | $e_{5}$ | $e_{4}$ | $-e_{6}$ | $e_{3}$ | $e_{2}$ | $-e_{0}$ | $-1$ | $e_{1}$
$e_{6}$ | $e_{6}$ | $e_{2}$ | $e_{5}$ | $-e_{6}$ | $e_{4}$ | $-e_{3}$ | $-e_{1}$ | $-1$

###  1.4. Overview of the article

In 1923 Dickson generalized the procedure of obtaining $\mathbb{C}$ from
$\mathbb{R}$, $\mathbb{H}$ from $\mathbb{C}$ and $\mathbb{O}$ from
$\mathbb{H}$ into what is now known as the ‘Cayley-Dickson Procedure’, a
recipe of creating a family of $2^{n}$-dimensional algebras over a given field
$F$ of characteristic $\neq 2$. The eight-dimensional members of this family
are called _Cayley algebras_ and share with $\mathbb{O}$ the important
property of being alternative (cf Cor. 1.3 above). Also the decomposition
$\mathcal{O}=F1\oplus\operatorname{Im}(\mathcal{O})$ makes sense over these
algebras (as well as their lower-dimensional counterparts) and behaves largely
the same as in the real cases described above. However not all Cayley algebras
are division algebras. Cayley algebras that contain zero-divisors are called
_split_.

Section 2 covers the preliminaries from non-associative algebra, focussing
towards the structure of the 1, 2, 4 and 8-dimensional algebras generated in
the Cayley-Dickson process over arbitrary fields of characteristic not 2. In
Section 3 we then prove our first main result:

######  Theorem (3.4).

Let $\mathcal{O}$ be a Cayley division algebra (i.e. a Cayley algebra that is
also a division algebra) over $F$ and let $V=\operatorname{Im}(\mathcal{O})$.
Let $\mathcal{H}$ be the collection of four-dimensional associative
subalgebras of $\mathcal{O}$ and
$\mathcal{B}\coloneqq\\{\operatorname{Im}(H)\colon H\in\mathcal{H}\\}$. Then
the pair $(V,\mathcal{B})$ is a $q$-Fano plane over $F$.

Theorem 3.4 generalizes Theorem 1.5 and hence it is not surprising that it is
derived from a more general version of Proposition 1.2 above:

######  Proposition (3.3).

In the setting of Thm 3.4 above, every two-dimensional subspace of $V$
generates a four-dimensional associative subalgebra of $\mathcal{O}$.

Section 4 focusses on understanding the role of the requirement that
$\mathcal{O}$ is a division algebra in the statement and proof of Proposition
3.3, by studying what would happen in case $\mathcal{O}$ is split. To this end
we divise a classification of all two-dimensional subspaces of
$\operatorname{Im}(\mathcal{O})$ into six types, based on the properties of
the subalgebras they generate. Here we are greatly helped by the fact that,
quite contrary to the famous dictum by Tolstoy, each Cayley division algebra
is a divsion algebra in its own way, but all split Cayley algebras are alike.
‘Alike’ here means concretely that not only are all split Cayley algebras over
a given field $F$ isomorphic (Theorem 2.28), but even stronger: they can all
be obtained by tensoring $F$ over $\mathbb{Z}$ with the same non-associative
ring (Lemma 4.4). Based on our classification of two-dimensional subspaces we
conclude that, with notation as in Thm. 3.4 above, no subset of $\mathcal{B}$
can be a $q$-Fano plane on $V$ in case $\mathcal{O}$ contains zero divisors
(Corrolary 4.23).

In Section 5 we then put a positive spin on the results of the previous
section as we prove the second main result of the paper:

######  Theorem (5.8).

Let $\mathcal{O}$ be a Cayley algebra over a field $F$ of characteristic $\neq
2$ and let $V=\operatorname{Im}(\mathcal{O})$. Let $\mathcal{H}$ be the
collection of four-dimensional subalgebras of $\mathcal{O}$ and
$\mathcal{B}\coloneqq\\{\operatorname{Im}(H)\colon H\in\mathcal{H}\\}$. Then
the pair $(V,\mathcal{B})$ is a $q$-covering design over $F$ with parameters
$(7,3,2)$, which is inclusion-minimal in the sense that no proper subset of
$\mathcal{B}$ is has the $q$-covering design property. If moreover
$\mathcal{O}$ is a division algebra, then $(V,\mathcal{B})$ is a $q$-Fano
plane over $F$.

The next three sections are dedicated to reformulating Theorem 5.8 into more
beautiful forms. To see what we mean by that, we return to the appearance of
the classical Fano-plane inside the real octonions $\mathbb{O}$. Having
settled on taking the basis elements $e_{0},\ldots,e_{6}$ as the point set of
our Fano plane, Theorem 5.8 suggests a way of describing the collection of
lines: a triple $\\{e_{i},e_{j},e_{k}\\}$ forms a line if and only if the
eight element set $\\{\pm 1,\pm e_{i},\pm e_{j},\pm e_{k}\\}$ is closed under
multiplication. Indeed it is easy to verify from the definition of octonion
multiplication in Section 1.3.2 that this reproduces the same collection of
lines we met in that section. However this description of the Fano plane is
also a bit ugly, involving the 9 additional elements $\pm
1,-e_{0},\ldots,-e_{6}$ that are not needed as points in the Fano-plane
itself. In Section 1.3.2 we met a much prettier description of the line set:
$\\{e_{i},e_{j},e_{k}\\}$ is a line in the Fano plane if and only if
$(e_{i}e_{j})e_{k}=e_{i}(e_{j}e_{k})$. In Section 6 we show that this
description of the Fano-plane has a $q$-analog as well: the set $\mathcal{B}$
of Theorem 5.8 can equivalently be described as

$\displaystyle\mathcal{B}$
$\displaystyle=\\{B\in\operatorname{Gr}_{3}(V)\colon(ab)c=a(bc)\textrm{ for
all }a,b,c\in B\\}$ $\displaystyle=\\{\operatorname{span}(a,b,c)\colon
a,b,c\in V,(ab)c=a(bc)\\}$

Here the Grassmanian $\operatorname{Gr}_{3}(V)$ is the set of all three-
dimensional subspaces of $V$.

As a (non-obvious) consequence of this equality we moreover derive (in Section
7) that the set $\mathcal{B}$ allows a more geometric description: embedding
$\operatorname{Gr}_{3}(V)$ into $\mathbb{P}(W)$ where
$W\coloneqq\bigwedge^{3}V$ by the Plücker embedding $\psi$, we find that the
subset $\mathcal{B}$ is described as the intersection of
$\psi(\operatorname{Gr}_{3}(V))$ with the image under the projection map of a
28-dimensional _linear_ subspace $K$ of the 35-dimensional space $W$.

In Section 8 we then set out to describe this space $K$ in entirely
combinatorial terms. The result is a slightly more general version of Theorem
1.1 above (Theorem 8.1) that provides a recipe for creating a
$(7,3,2)$-$q$-covering design over a given field $F$ (with
$\operatorname{char}(F)\neq 2$) from the classical Fano plane in an ‘algebra-
free’ way. The motivation for including such a result is the hope that upon
reading it someone might ‘guess’ the correct way to create $q$-versions of
Steiner triple systems at different parameters, where the underlying algebra
is not available. Needless to say, the proof of such a result would have to be
quite different from the proof of Theorem 8.1 (which relies on Thm 5.8 above)
and we won’t speculate on it here.

Although similar in form, Thm 8.1 is more general than Theorem 1.1 above,
encompassing among others the $q$-Fano planes (not covered by Thm 1.1 but
mentioned just below it) over the function fields $F=K(\alpha,\beta,\gamma)$
of transedence degree three over any field $K$ with
$\operatorname{char}(K)\neq 2$. All statements in Theorem 1.1 above follow as
straightforward special cases of Theorem 8.1 except for the last, more
quantative part, where it states that the number of blocks in the $q$-covering
design over a field with $q$ elements is
$q^{8}+q^{7}+2q^{6}+2q^{5}+3q^{4}+2q^{3}+2q^{2}+q+1$.

The verification of this number is the content of the final section of the
paper, Section 9. Assisted by the classic fact (discussed in Section 2) that
all Cayley algebras over a finite field contain zero divisors and hence are
split, we use our classification of Section 4 to count the number of four-
dimensional subalgebras of such an algebra and hence the minimum number of
blocks in a $(7,3,2)$-$q$-covering design over a finite field obtainable by
our method.

###  1.5. Acknowledgements

Section 2 of the current paper is loosely based on Chapters 1 and 2 of my
(unpublished) master’s thesis [Noo05]. Although the objective of that thesis
was perpendicular to that of the current paper (defining $q$-analogs of the
octonions rather than using the octonions to define $q$-analogs of something
else) it was the research done for that project that enabled me, more than a
decade later, to recognize the truth of Theorem 1.5 upon learning the
definition of a $q$-Fano plane. I’d like to thank my thesis advisors, Tom
Koornwinder and Eric Opdam for suggesting the topic and for their guidance in
those days.

Most of all I’d like to thank Relinde Jurrius for introducing me to the
notions of $q$-Fano plane and $q$-covering design, for careful comments on an
earlier version of this paper and for being the go-to expert on all things
related to finite geometry over the years.

##  2\. Preliminaries from algebra

###  2.1. Conventions and definitions

######  Convention 2.1.

Throughout the paper, let $F$ be a field of characteristic unequal to 2. We
will denote elements of $F$ (scalars) by lower case Greek letters, vector
spaces over $F$ will be denoted by capital roman letters and elements of such
vector spaces will be denoted by lower case roman letters.

######  Definition 2.2.

An _algebra_ over $F$ is a vector space $A$ over $F$ equiped with an bilinear
multiplication map $A\times A\to A$. An algebra $A$ is called _unital_ if
there exists an element $1\in F$ such that $1a=a1=a$ for all $a\in A$. It is
called associative when $(ab)c=a(bc)$ for all $a,b,c$ in $A$.

######  Convention 2.3.

Throughout this paper, by an _algebra_ we will denote a finite-dimensional,
unital, but not necessarily associative algebra over $F$.

Let $A$ be an algebra. The unit element of $A$ will be denoted $1$ and the
one-dimensional subalgebra of scalar multiples of 1 will be denoted $F1$. By
bilinearity of the product on $A$ we have that all elements of $F1$ commute
and associate with all elements of $A$.

######  Notation 2.4.

Let $A$ be an algebra. For any subset $B$ of $A$ we denote by $\langle
B\rangle$ the subalgebra of $A$ generated by the elements of $B$. That is: the
intersection of all subalgebras of $A$ containing $B$. When $B$ is finite,
e.g. $B=\\{u,v\\}$, we will write $\langle u,v\rangle$ for
$\langle\\{u,v\\}\rangle$.

######  Convention 2.5.

Since we required all our algebras to be unital, we demand the same of their
subalgebras. In particular we have that $F1\subseteq\langle B\rangle$ for any
subset $B$ of $A$.

The following elementary observation is stated explicitly, not because I think
the reader is stupid but because I woke up one night in panick, believing that
there were a hole in the proof amounting to the following fact being false.
However it is true and promoted to a lemma so that I can refer back to it
later.

######  Lemma 2.6.

Let $A$ be an algebra, let $U\subseteq A$ be a linear subspace and let
$u_{1},\ldots,u_{k}$ be a basis of $U$. Then $\langle
u_{1},\ldots,u_{k}\rangle=\langle U\rangle$.

###### Proof.

The inclusion $\langle u_{1},\ldots,u_{k}\rangle\subseteq\langle U\rangle$ is
obvious from the inclusion $\\{u_{1},\ldots,u_{k}\\}\subseteq U$. For the
converse inclusion we note that every algebra containing
$\\{u_{1},\ldots,u_{k}\\}$ contains $U$ and hence so does the intersection
$\langle u_{1},\ldots,u_{k}\rangle$ of all such algebras. But this means that
$\langle u_{1},\ldots,u_{k}\rangle$ is an algebra containing $U$ and hence it
certainly contains the intersection of all such algebras, that is $\langle
u_{1},\ldots,u_{k}\rangle\supseteq\langle U\rangle$. ∎

######  Definition 2.7.

An algebra is _alternative_ if every subalgebra generated by two elements is
associative.

Alternative algebras need not be associative, but many results that are
familiar for associative algebras do hold for more general alternative
algebras as well. We will see an example below in Lemma 2.10, many more
examples can be found in the book by Schafer [Sch66].

######  Definition 2.8.

We say that an algebra $A$ is a _division algebra_ if for every non-zero $a\in
A$ the linear maps $L_{a}\colon x\mapsto ax$ and $R_{a}\colon x\mapsto xa$ are
invertible in $\operatorname{End}(A)$.

######  Definition 2.9.

A non-zero element $x$ in an algebra $A$ is called a _zero-divisor_ if there
exist non-zero $y\in A$ such that $xy=0$ or $yx=0$.

######  Lemma 2.10.

Let $A$ be a finite-dimensional alternative algebra. Then the following are
equivalent:

  1. (1)

$A$ is a division algebra

  2. (2)

$A$ has no zero-divisors

  3. (3)

For every non-zero $a\in A$ there exist an element $a^{-1}\in A$ such that
$aa^{-1}=a^{-1}a=1$.

The proof is left as an exercise to the reader. A counterexample to the
implications $2\implies 1$ and $2\implies 3$ when we drop the condition that
$A$ is finite-dimensional is given by the algebra $F[x]$ of polynomials over
$F$. A counterexample to the implication $3\implies 2$ when we drop the
condition that $A$ is alternative is given in Section 2.5.

######  Definition 2.11.

An algebra is _quadratic_ if the subalgebra generated by the single element
$a$ is two-dimensional for all $a\not\in F1$.

######  Definition 2.12.

By an _involution_ $*$ of an algebra $A$ we denote a linear anti-automorphism
of order dividing 2. In other words and involution is a linear map $a\mapsto
a^{*}$ satisfying

(4) $(ab)^{*}=b^{*}a^{*}\qquad(a^{*})^{*}=a$

for all $a,b\in A$.

By linearity $\alpha^{*}=\alpha$ for all $\alpha\in F1$.

######  Definition 2.13.

We say that an involution $*$ is a _strong involution_ if the converse to the
above holds, i.e. if $a^{*}=a$ if and only if $a\in F1$.

The most famous example of a strong involution is complex conjugation in the
algebra $A=\mathbb{C}$ over $F=\mathbb{R}$. The quaternion conjugation (3) on
the $\mathbb{R}$-algebra $\mathbb{H}$ discussed in the Introduction is another
example of a strong involution. Moreover $\mathbb{C}$ and $\mathbb{H}$ are
both examples of quadratic algebras over $\mathbb{R}$. We will see in Section
2.3 that this is no coincidence.

###  2.2. A theorem of Artin

We defined an alternative algebra as an algebra in which the subalgebra
generated by any two elements is associative. Alternativity can be viewed as a
‘controlled’ form of non-associativity, where we bear in mind that all
associative algebras are automatically alternative. Clearly for any $a,b$ in
an alternative algebra we have that

(5) $\displaystyle(aa)b=a(ab)$ (6) $\displaystyle(ab)a=a(ba)$ (7)
$\displaystyle(ba)a=b(aa)$

It is an interesting result of Artin that the converse holds:

######  Theorem 2.14 ([Sch66], Thm III.3.1).

Let $A$ be an algebra. The following are equivalent

  1. (1)

$A$ is alternative

  2. (2)

At least two of the three equations (5, 6, 7) hold for all $a,b\in A$

  3. (3)

The _associator_ , the trilinear map $(.,.,.)\colon A\times A\times A\to A$
defined by $(a,b,c)=(ab)c-a(bc)$ _alternates_ , that is: changes sign under
odd permutations of its entries.

It is the third item that explains the name alternative. Note that $(a,b,c)=0$
for all $a,b,c$ when $A$ is associative, thus providing a second way of seeing
that all associative algebras are alternative.

###  2.3. A particularly nice class of algebras

Let $A$ be a quadratic algebra and $a\in A,a\not\in F1$. By definition of
quadratic algebra there are unique scalars $\tau(a),n(a)\in F$ such that

(8) $a^{2}=2\tau(a)a-n(a)$

This defines maps $\tau,n$ from $A\backslash F1\to F$ and we extend these maps
to all of $A$ by setting $\tau(\alpha)=\alpha,n(\alpha)=\alpha^{2}$ for
$\alpha\in F1$. Note that with this convention equation (8) holds for all
$a\in A$.

It is an easy but important observation that the map $\tau\colon A\to F$ is
linear. Moreover, and this is the reason for the appearance of the number 2 in
(8), we have that $\tau$ is a linear projection operator onto the space $F1$,
that is, we have that $\tau\circ\tau=\operatorname{Id}$. When $F=\mathbb{R}$
we think of $\tau$ as ‘taking the real part of an element’.

We define $\operatorname{Im}(A)=\ker\tau$. Since $\tau$ is a projection
operator we obtain a decompostion (of linear spaces, not of algebras):

(9) $A=\tau(A)\oplus\operatorname{Im}(A).$

The elements of the co-dimension-1 linear space $\operatorname{Im}(A)$ are
called _imaginary_. The reason for this terminology stems from the special
case $F=\mathbb{R},A=\mathbb{C}$ and becomes apparent from the following
alternative characterization of $\operatorname{Im}(A)$ (the equivalence of the
two characterizations dates back to Frobenius and follows easily from (8)):

(10) $\operatorname{Im}(A)=\\{a\in A\colon a^{2}\in F1\textrm{ but }a\not\in
F1\\}\cup\\{0\\}.$

We will be interested in quadratic algebras satisfying the additional
condition

(11) $\tau(ab)=\tau(ba)\textnormal{ for all }a,b\in A.$

We have:

######  Theorem 2.15.

Let $A$ be an algebra. The following are equivalent:

  1. (1)

$A$ is a quadratic algebra satisfying (11),

  2. (2)

$A$ possesses a strong involution.

###### Proof.

‘$1\implies 2$’: Given the quadratic algebra structure we write $\tau$ for the
map $a\mapsto\tau(a)1$ (in other words, we reinterpret $\tau$ as mapping into
the subalgebra $F1$ of $A$ rather than into the ‘external’ field $F$.) We
write $\operatorname{Im}=\operatorname{Id}-\tau$ for the linear projection
onto the space $\operatorname{Im}(A)$ along the decompostion (9).

The involution $*$ is then given by

(12) $a^{*}=\tau(a)-\operatorname{Im}(a)$

Let $u,v\in\operatorname{Im}(A)$. From (10) we have that $(u+v)^{2}$, $u^{2}$,
and $v^{2}$ are all in $F1$ so that from expanding $(u+v)^{2}$ we find that
$uv+vu\in F1$ as well. It follows that $\tau(uv+vu)=uv+vu$ and hence
$\operatorname{Im}(uv+vu)=0$ so that
$\operatorname{Im}(uv)=-\operatorname{Im}(vu)$.

Combining this with (11) and (12) yields
$(uv)^{*}=\tau(uv)-\operatorname{Im}(uv)=\tau(vu)+\operatorname{Im}(vu)=vu$.
Now since $u,v\in\operatorname{Im}(A)$, $v^{*}=-v$ and $u^{*}=-u$ by (12) and
hence the equation $(uv)^{*}=vu$ extends to $(uv)^{*}=v^{*}u^{*}$, showing
that for $u,v\in\operatorname{Im}(A)$ the first relation in (4) holds. Writing
$a=\alpha+u$, $b=\beta+v$ with $\alpha,\beta\in F1,u,v\in\operatorname{Im}(A)$
we can easily extend this to general $a,b\in A$ exploiting the linearity of
the $*$-operator.

The second equation in (4) is obvious from (12). This shows that ∗ is an
involution; the fact that it is strong follows directly from (12).

‘$2\implies 1$’: Given a strong involution $*$ we define the maps $\tau$ and
$n$ by

(13) $\tau(a)=\frac{1}{2}(a+a^{*})$

and

(14) $n(a)=a^{*}a$

Note that the strongness of the involution guarantees that $\tau$ and $n$ take
values inside $F1$, allowing us to reinterpret them as taking values in $F$.
It is then straightforward to verify from (4) that (8) and (11) hold. (8) in
turn implies that $A$ is quadratic. ∎

Theorem 2.15 provides a third characterization of $\operatorname{Im}(A)$ for
algebras $A$ with a strong involution:

(15) $\operatorname{Im}(A)=\\{a\in A\colon a^{*}=-a\\}$

In other words: $\operatorname{Im}(A)$ is the $(-1)$-eigenspace of the strong
involution. Since by definition of ‘strongness’ the $1$-eigenspace is one-
dimensional and since linear transformations of order $\leq 2$ cannot have
proper Jordan blocks, we obtain a second proof of a fact that we will now give
its own number:

######  Lemma 2.16.

The co-dimension of $\operatorname{Im}(A)$ in an algebra $A$ equipped with a
strong involution equals 1.

A very useful property of imaginary elements is

(16) $uv=-vu\qquad\textrm{ for all }u,v\in\operatorname{Im}(A)$

Which follows from combining (15) with (4).

Finally we note that from (14) and (4) if follows that in alternative algebras
$A$ with a strong involution the function $n$ is multiplicative:

(17) $n(ab)=n(a)n(b)\qquad\textnormal{for all }a,b\in A.$

For the $(F=\mathbb{R})$-algebras discussed in the introduction we have that
the function $n$ equals the square of the Euclidean norm on $A$ and the proof
of (17) is spelled out in (2).

###  2.4. The Dickson Double

######  Definition 2.17.

Let $A$ be an $F$-algebra with a strong involution $*$ and let $\gamma\in
F\backslash\\{0\\}$ be a non-zero element of $F$. The _Dickson double_
$D_{\gamma}(A)$ of $A$ is an algebra with dimension twice the dimension of $A$
formed by attaching to $A$ an element $i$ satisfying

(18) $i^{2}=\gamma.$

In more detail we have that as a vector space $D_{\gamma}(A)$ equals $A\oplus
iA$ where elements of the subspace $iA$ are interpreted as the product of the
‘new’ element $i=i1$ with elements of $A$. Multiplication is defined by the
formula

(19) $(a+ib)(c+id)=(ac+\gamma db^{*})+i(a^{*}d+cb)$

The involution $*$ on $A$ is extended to $D_{\gamma}(A)$ as the unique
involution agreeing with the original $*$ on the subalgebra $A$ and satisfying
$i^{*}=-i$.

######  Lemma 2.18.

The multiplication (19) can be equivalently be defined by the three equations

(20) $\displaystyle p(iq)=i(p^{*}q)$ (21) $\displaystyle(pi)q=(pq^{*})i$ (22)
$\displaystyle(ip)(qi)=\gamma(pq)^{*}$

for all $p,q\in A$ or by the single equation

(23) $(a+bi)(c+di)=(ac+\gamma d^{*}b)+(da+bc^{*})i.$

Particularly useful in computations is also the corollary

(24) $xi=ix^{*}\textnormal{ for all }x\in A$

from which in turn it follow that

(25) $(xi)^{*}=-(xi)\textnormal{ for all }x\in A$

From (25) we note:

######  Corollary 2.19.

The involution $*$ on $D_{\gamma}(A)$ is strong.

We’ll use the last corollary to try and avoid doing computations using (19 \-
25) in this paper and instead rely on the properties of algebras with a strong
involution derived in Section 2.3.

######  Remark 2.20.

One of the things apparent from equations (20, 21, 22) is that we cannot
assume the Dickson double $D$ of an algebra $A$ to be associative, even if $A$
is. On the other hand it is easy to see that if we happen to know that $A$ is
a subalgebra of an associative algebra $D$ and $i$ is an element of $D$ not
contained in $A$ then (24), if true in $D$, implies equations (20, 21, 22) to
hold in $D$ as well.

The following theorem, due to Albert, (which can be derived directly from
equations (20, 21, 22) in conjunction with Artin’s theorem above) states
however that for $A$ of dimension 1, 2 or 4 (and $D$ subsequently of dimension
2, 4 or 8) the non-associativity is controlled.

######  Theorem 2.21.

Let $A$ be a an $F$-algebra with a strong involution and let $D$ be its
Dickson double. Then:

$D$ is commutative if and only if $A=F$.

$D$ is associative if and only if $A$ is commutative.

$D$ is alternative if and only if $A$ is associative.

Now that alternativity is on the table it is interesting to know that Remark
2.20 in fact generalizes to the alternative case though with a much more
complicated proof:

######  Lemma 2.22.

Let $D$ be an alternative algebra with strong involution $*$, $A$ a subalgebra
of $D$ that is closed under application of $*$, and let
$i\in\operatorname{Im}(D)$ be an element _not_ contained in $A$ satisfying
(24). Define $\gamma\in F$ by $i^{2}=\gamma 1$. Then
$(\ref{pq1},\ref{pq2},\ref{pq3})$ hold in $D$ for all $p,q\in A$. It follows
that if $\gamma\neq 0$ then $D$ contains a copy of the Dickson double
$D_{\gamma}(A)$ of $A$.

###### Proof.

We’ll start by writing out the proof of (21).

$(pi)q=p(iq)+(p,i,q)=p(q^{*}i)+(p,i,q)=(pq^{*})i-(p,q^{*},i)+(p,i,q)$

where the first and third equality apply the definition of the associator
$(.,.,.)$ given in Theorem 2.14 and the second equality applies (24). Hence
equation (21) follows as soon as we show that $(p,i,q)-(p,q^{*},i)=0$. But by
alternativity and trilineairty of the associator (Thm 2.14) we find that

$(p,i,q)-(p,q^{*},i)=(p,i,q)+(p,i,q^{*})=(p,i,q+q^{*})=(p,i,2\tau(q)1)$

where the last equality uses (13). And since elements of $F1$ associate with
all elements of $D$ we find that $(p,i,2\tau(q)1)=0$ as desired.

Equality (20 can be derived from (21) and (24) as follows. By (24) and (4) we
see that the right hand side of (20) equals $(q^{*}p)i$. Applying (21) with
$q^{*}$ in the role of $p$ and $p^{*}$ in the role of $q$ we obtain that this
equals $(q^{*}i)p^{*}$. We have thus obtained the equality
$(pq^{*})i=(q^{*}i)p^{*}$.

Applying the $*$-operator to both sides of this last equality we find on te
left hand side $-(pq^{*})i$ by (25) and on the right hand side $p(i^{*}q)$ by
applying (4) both inside and outside the brackets. Since $i^{*}=-i$ the right
hand side simplifies to $-p(iq)$ and multiplying both sides with $-1$ we
finally obtain $(pq^{*})i=p(iq)$, which is (20).

Finally to see that (22) holds as well we first establish that (24) also holds
with $qi$ in the role of $i$. Concretely, for any $x\in A$ we have

(26)
$x(qi)=\tau(x)(qi)+\operatorname{Im}(x)(qi)=(qi)\tau(x)-(qi)\operatorname{Im}(x)=(qi)x^{*}.$

Here the second equality uses (16) and the fact that elements of $F1$ commute
with everything and the third equality uses (13).

Now we use this to attack (22). Using the definition of the associator for the
first and third equality and (26) for the second we obtain:

(27)
$(ip)(qi)=i(p(qi))+(i,p,qi)=i((qi)p^{*})+(i,p,qi)=(i(qi))p^{*}-(i,qi,p^{*})+(i,p,qi).$

As above we find that $-(i,qi,p^{*})+(i,p,qi)=(i,2\tau(p),qi)=0$ so that (27)
reduces to

(28) $(ip)(qi)=(i(qi))p^{*}.$

The left hand side of (28) equals the left hand side of (20). We look at the
term $i(qi)$ within the outermost brackets of the right hand side of (28).
From (24) we see that it equals $i(iq^{*})$ and since this term lives in the
associative subalgebra generated by the elements $i$ and $q$ we can shift the
brackets and conclude that it equals $i^{2}q^{*}=\gamma q^{*}$. The right hand
side of (28) then becomes $\gamma q^{*}p^{*}$ which needs no brackets as
$\gamma\in F$. By (4) this equals the right hand side of (22). ∎

###  2.5. The Cayley-Dickson algebras

We inductively define the class of Cayley-Dickson algebras. Each such algebra
is a finite-dimensional $F$-algebra with a strong involution.

######  Definition 2.23.

The (unique) one-dimensional $F$-algebra $F$ is a Cayley-Dickson algebra where
the strong involution is the identity. An $n$-dimensional $F$-algebra $A$ for
$n>1$ is a Cayley-Dickson algebra if it is the Dickson Double of a Cayley-
Dickson algebra $B$ of lower dimension.

It follows that every Cayley-Dickson algebra has dimension equal to a power of
2.

It also follows from Theorem 2.15 that all Cayley-Dickson algebras $A$ are
quadratic algebras with a co-dimension 1 subspace $\operatorname{Im}(A)$ of
imaginary elements $u$ satisfying $u^{2}\in F1$, $u^{*}=-u$. We note the
similarity to equations (18) and (25) implying that the ‘special’ element $i$
used in the construction of $A$ from the smaller algebra $B$ is always
contained in the space $\operatorname{Im}(A)$.

######  Remark 2.24.

In fact the element $i$ and subalgebra $B$ are not _that_ special. Once we are
presented with a Cayley-Dickson algebra $A$ of dimension $2^{n}\leq 8$, it is
impossible to tell which of its many $2^{n-1}$-dimensional Cayley-Dickson
subalgebras $B$ was used to construct $A$, in the sense that $A$ can be
realized as a Dickson double of each of them. Various choices of the
subalgebra $B$ will allow various elements $u\in\operatorname{Im}(A)$ with
$u^{2}\neq 0$ to play the role of the ‘special’ element $i$. We will split the
proof this remarkable fact into two parts. The proof in case that $A$ is a
division algebra is presented here (Prop. 2.25 below); the remaining cases
will be dealt with in Theorem 6.8. A different proof of the division algebra
case in the special case that $\dim A=4$ appears as part of the proof of
Proposition 3.3.

######  Proposition 2.25.

Let $A$ be a Cayley-Dickson division algebra of dimension $2^{n}$ for
$n\in\\{1,2,3\\}$ and let $B$ be a $2^{n-1}$-dimensional subalgebra of $A$
closed under the action of the strong involution. Then there exists a
$\gamma\in F^{\times}$ and $i\in\operatorname{Im}(A)$, not contained in $B$
such that $A=B\oplus iB$ as a vector space and such that
$(\ref{pq1},\ref{pq2},\ref{pq3})$ hold in $A$ for all $p,q\in B$.
Consequently, $A$ is isomorphic to the Dickson double $D_{\gamma}(B)$.

###### Proof.

Let $\\{b_{1},\ldots,b_{2^{n-1}}\\}$ be a basis of $B$. Let the linear maps
$R_{b_{j}}$ be defined as in Lemma 2.10. Since $\ker\tau$ has co-dimension 1
(Lemma 2.16), for each $j\in\\{1,\ldots,2^{n-1}\\}$ the space $\ker(\tau\circ
R_{b_{j}})$ has co-dimension at most 1. (In fact these co-dimensions are
exactly 1 by Lemma 2.10.) It follows that the space $\bigcap_{j=1}^{\dim
B}\ker(\tau\circ R_{b_{j}})$ of elements $i$ such that
$ib\in\operatorname{Im}(A)$ for every $b\in B$ has co-dimension at most
$2^{n-1}$ and hence in particular is non-zero. Pick any $i\neq 0$ from this
space. We note that $i\in\operatorname{Im}(A)$ and hence $i^{2}=\gamma 1$ for
some $\gamma\in F$ where moreover $\gamma\neq 0$ since otherwise $i$ would be
a divisor of zero.

Now since $A$ is a division algebra, $L_{i}$ is invertible (Lemma 2.10) and
hence the space $iB$ is $2^{n-1}$-dimensional. We also have that $iB\cap
B=\\{0\\}$ since if $ia=b$ for some $a,b\in B$ with $a\neq 0$ it follows that
$i=ba^{-1}\in B$, and hence $i^{-1}=\frac{1}{\gamma}i\in B$. But this would
imply that $1=ii^{-1}\in\operatorname{Im}(A)$ by our choice of $i$ – a clear
contradiction. We conclude that $B+iB$ is $2^{n}$-dimensional and hence equal
to all of $A$.

It remains to verify $i$ and $B$ satisfy (20, 21, 22). For this we first note
that $i$ satisfies (25) for all $x\in B$ since, by our construction of $i$,
$ix\in\operatorname{Im}(A)$ for each $x\in B$. From (25) we then conclude that
(24) holds for all $x\in B$ as well and then deduce (20, 21, 22) from (24) as
in Lemma 2.22. In this last step we exploit that, as $\dim(A)\leq 8$ we have
that $A$ is alternative by Theorem 2.21. ∎

######  Definition 2.26.

A Cayley-Dickson algebra of dimension 8 is called a _Cayley algebra_ or an
_octonion algebra_. A Cayley-Dickson algebra of dimension 4 is called a
_quaternion algebra_.

By Theorem 2.21, all Cayley algebras are alternative and all quaternion
algebras are associative. In the literature on associative algebras a
different, yet equivalent, definition of the term quaternion algebra is used,
we’ll come back to that issue in Section 2.8.

By the same theorem, all two-dimensional Cayley-Dickson algebras are
commutative. In particular, when they are division algebras, they are fields.
And, as they are formed by attaching to $F$ a new squareroot (the element $i$
from the definition) of some $\alpha\in F$, they are quadratic field
extensions of $F$. In particular:

######  Lemma 2.27.

Let $\alpha\in F^{\times}$. The two-dimensional Cayley-Dickson algebra
$D_{\alpha}(F)$ is a quadratic field extension of $F$ if and only if $\alpha$
is a non-square in $F$ and contains divisors of zero if and only if $\alpha$
is a square in $F$.

###### Proof.

We only need to prove the ‘if’ directions. In both cases we tacitly exploit
that $D_{\alpha}(F)$ is commutative by Theorem 2.21. If $\alpha$ is a non-
square the polynomial $x^{2}-\alpha$ is irreducible over $F$ and attaching the
root $i$ of this polynomial to $F$ yields a quadratic field extension by the
standard argument. On the other hand when $\alpha=\beta^{2}$ for some
$\beta\in F$ we find that $(\beta-i)(\beta+i)=0$ by the ‘strange product’
formula. ∎

The following surprising fact forms the basis underlying all our
considerations from Section 4 onwards.

######  Theorem 2.28 (Uniqueness theorem, [Sch66], p. 24-27).

In each of the dimensions 2, 4, 8 there is up to isomorphism exactly one
Cayley-Dickson algebra containing divisors of zero, called the _split_ Cayley
Dickson algebra in that dimension.

The number of alternative Cayley-Dickson algebras _not_ containing divisors of
zero (hence division algebras by Lemma 2.10) depends on the field $F$. Cayley-
Dickson algebras of dimension 16 and greater (equivalently: non-alternative
Cayley-Dickson algebras) will be of no concern of us, although we note that
the 16-dimensional $\mathbb{R}$-algebra
$D_{-1}(D_{-1}(D_{-1}(D_{-1}(\mathbb{R}))))$ (the sedenions) are interesting
as an example of an algebra containing zero-divisors while at the same time
every non-zero element has a multiplicative inverse (cf Lemma 2.10).

We collect some consequences of the uniqueness theorem.

######  Corollary 2.29.

An alternative Cayley-Dickson algebra is split if and only if it contains an
imaginary square root of 1

###### Proof.

For the ‘if’ direction, let $e$ be a imaginary square-root of 1. Then
$(1+e)\neq 0,(1-e)\neq 0,(1+e)(1-e)=0$, showing the existence of zero-
divisors. Conversely: $D_{1}(F)$ and hence $D_{\beta}(D_{1}(F))$ and
$D_{\alpha}(D_{\beta}(D_{1}(F)))$ contain an imaginary square-root of 1 by
construction and hence the ‘only if’ direction follows from the uniqueness
theorem. ∎

We can however be a bit more explicit.

######  Proposition 2.30.

The (by Theorem 2.28) unique two-dimensional split Cayley-Dickson algebra is
isomorphic to the algebra $F\oplus F$ of pairs of elements in $F$ with
pointwise addition and multiplication. Here the $*$ operation is given by
$(\alpha,\beta)^{*}=(\beta,\alpha)$

###### Proof.

Taking uniqueness for granted we only need to show that this algebra is
isomorphic to $D_{1}(F)$. Let $i$ be the element of that algebra used in the
definition of the doubling process (so in this case $i^{2}=1$). Then
$\\{1,i\\}$ is a basis of $D_{1}(F)$ and the isomorphism is given by
$1\mapsto(1,1)$, $i\mapsto(1,-1)$. ∎

######  Proposition 2.31.

With $\tau,n,*$ as in Section 2.3, the (by Theorem 2.28 unique) four-
dimensional Cayley-Dickson algebra over $F$ containing divisors of zero is
isomorphic to the matrix algebra $\operatorname{Mat}(2,F)$ with $1=I$,
$\tau(X)=\frac{1}{2}\operatorname{Tr}(X)$, $n(x)=\det(X)$ and
$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}^{*}=\begin{pmatrix}d&-b\\\
-c&a\end{pmatrix}$.

This relation to the matrix trace is also the reason that the operator $\tau$
is called $\tau$.

###### Proof.

Taking uniqueness for granted, we only need to show an isomorphism between the
matrix algebra and $D_{1}(D_{-1}(F))$. It is easy to verify that the
subalgebra $\\{\begin{pmatrix}a&-b\\\ b&a\end{pmatrix}\colon a,b\in F\\}$ of
$\operatorname{Mat}(2,F)$ is isomorphic to $D_{-1}(F)$ with the involution
being the restriction to this subalgebra of the the involution described in
the proposition. Denoting this subalgebra by $B$ for the moment, it takes only
a little more effort to verify that the algebra $\operatorname{Mat}(2,F)$ is
isomorphic to the Dickson Double $D_{1}(B)$ of $B$ where the role of the
element $i$ (the ‘new square-root of 1’) is played by the matrix
$i=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$. ∎

In a sense the split Cayley and quaternion algebras are the richest Cayley and
quaternion algebras:

######  Corollary 2.32.

Every isomorphism class of Cayley-Dickson algebras of dimension $2^{n-1}$
$(n=1,2,3)$ is represented among the subalgebras of the split Cayley-Dickson
algebra of dimension $2^{n}$.

###### Proof.

Given a $2^{n-1}$-dimensional Cayley-Dickson algebra $A$ we can realize the
split Cayley-Dickson algebra of dimension $2^{n}$ as $D_{1}(A)$. ∎

However, over certain important fields this richness is not too impressive, as
the following two corollaries show.

######  Corollary 2.33.

Every alternative Cayley-Dickson algebra over an algebraically closed field
$F$ is split.

###### Proof.

Let $u$ be a non-zero imaginary element in such an algebra. By (10), $u^{2}\in
F$ and by algebraic closedness there exists an $\alpha\in F$ such that
$\alpha^{2}=u^{2}$. If $u^{2}=0$ we have that $u$ is a zero-divisor and we are
done. Otherwise $(u/\alpha)$ is an imaginary square-root of 1 and we can apply
Corollary 2.29. ∎

######  Corollary 2.34.

Over a finite field $F$ there is up to isomorphism only one quaternion algebra
and only one Cayley algebra.

###### Proof.

By Wedderburn’s little theorem, every division algebra over $F$ is
commutative. By Theorem 2.21 this means that no 4-dimensional Cayley-Dickson
algebra can be a division algebra and hence the last proposition implies that
all 4-dimensional Cayley-Dickson algebras over $F$ are isomorphic to
$\operatorname{Mat}(2,F)$. Moreover, since every Cayley algebra $C$ over $F$
is obtained by doubling a quaternion algebra $Q$ over $F$, which it contains
as a subalgebra, any zero-divisor present in the quaternion algebra used to
obtain $C$ will also be an element of $C$. As we just saw that the presence of
zero-divisors in $Q$ is inevitable, the uniqueness theorem tells us that $C$
is unique up to isomorphism. ∎

To end this subsection on a happy note we recall that we already saw in the
introduction an example of a Cayley algebra that _is_ a division algebra, the
algebra $\mathbb{O}\coloneqq D_{-1}(D_{-1}(D_{-1}(\mathbb{R})))$ over
$F=\mathbb{R}$. The proof presented there that $\mathbb{O}$ is a division
algebra relied on the fact that the Euclidean norm $\|.\|$ on $\mathbb{R}^{8}$
respects the octonion multiplication: $\|x\|^{2}\|y\|^{2}=\|xy\|^{2}$. In the
context of more general Cayley-Dickson algebras this is just equation (17)
combined with the fact that in the special case of $\mathbb{O}$ the function
$n$ equals the square of the Euclidean norm. This suggests a path forward to
finding other fields over which a Cayley division algebra might exist.

######  Lemma 2.35.

Let $A$ be a Cayley-Dickson algebra. Then $A$ contains a zero divisor if and
only it contains a non-zero element $x$ such that $n(x)=0$.

###### Proof.

For the if-direction it is enough to recall that $xx^{*}=n(x)$ (equation
(14)). For the only-if-direction let $a,b\in A$ such that $ab=0$. We note that
by (8) we have that $n(0)=0$ so that the multiplicativity of $n$ (equation
(17)) implies that $n(a)=0$ or $n(b)=0$. ∎

Apparently the key to understanding the (non-)existence of zero divisors is
understanding the quadratic form $n\colon A\to F$. We’ll return to this in
Section 2.7.

###  2.6. Multiplication table and relation to the Fano plane

In Section 1 we gave a multiplication table (Table 1) of
$\mathbb{O}=D_{-1}(D_{-1}(D_{-1}(\mathbb{R})))$ with respect to the ‘standard’
basis. We also indicated how, when viewing the seven imaginary elements among
these standard basis elements as the points in a classical Fano plane, the
lines in that Fano plane can be used to describe the multiplication. In this
section we will provide the proof of those statements by providing a similar
table, Table 2, for the general Cayley algebra $\mathcal{O}\coloneqq
D_{\gamma}(D_{\beta}(D_{\alpha}(F)))$, where $F$ is any field and
$\alpha,\beta,\gamma$ are elements of $F^{\times}$. A special case of interest
(see also Section 2.7) occurs when $\alpha,\beta,\gamma$ are algebraically
independent, transcedental elements w. r. t. a subfield $K$ of $F$, but this
requirement is not necessary for the correctness of the table. In particular
we see that when choosing $\alpha=\beta=\gamma=-1$ in Table 2 we recover Table
1, showing that Table 1 is indeed the correct multiplication table, not only
of $\mathbb{O}=D_{-1}(D_{-1}(D_{-1}(\mathbb{R})))$ but for the Cayley algebras
$D_{-1}(D_{-1}(D_{-1}(F)))$ over any $F$.

Although Table 2 is a bit less transparant than Table 1, the underlying Fano
plane structure is still visible when igorning the scalars. In particular: for
$a,b,c\in\\{e_{0},\ldots,e_{6}\\}$ we have that $abc\in F1$ (for both
placements of the brackets) if and only if $\\{a,b,c\\}$ is a line in the
standard Fano plane structure on the $e_{i}$. Since moreover for _any_
imaginary elements $a,b\in\operatorname{Im}(\mathcal{O})$ with $a^{2}\neq 0$
we have that $ab\in F1$ if and only if $b\in Fa$, we find that the product of
two imaginary basis elements $e_{i},e_{j}$ is always a scalar multiple of the
third basis element $e_{k}$ lying on the line spanned by $e_{i},e_{j}$ in the
Fano plane structure on $\\{e_{0},\ldots,e_{6}\\}$.

The goal of this subsection is to prove these statements and compute the
mentioned ‘scalar multiples’, i.e. to verify the correctness of Table 2. In
other words, our goal is to recover the points and lines of the Fano plane,
together with the additional information in Table 2, from the Cayley algebra
$\mathcal{O}$. The opposite direction, obtaining the Cayley algebra (and
subsequent $q$-covering design) from the Fano plane together with some
additonal information as in Theorem 1.1, will be the subject of Section 8.

######  Notation 2.36.

For the duration of this subsection let
$\mathcal{O}=D_{\gamma}(D_{\beta}(D_{\alpha}(F)))$ and let
$e_{0},e_{1},e_{2}\in\operatorname{Im}(\mathcal{O})$ denote the elements $i$
used in the first, second and third Dickson-doubling step respectively. Hence
we have $e_{0}^{2}=\alpha 1$, $e_{1}^{2}=\beta 1$ and $e_{2}^{2}=\gamma 1$.

Altough the doubling process puts a clear ‘hierarchy’ on the elements
$e_{0},e_{1},e_{2}$, here we adopt a different perspective where the three
elements $e_{0},e_{1},e_{2}$ are treated on equal footing, as _generators_ of
$\mathcal{O}$; the remaining basis elements are then derived from them as
products: $e_{3}\coloneqq e_{0}e_{1}$, $e_{4}\coloneqq e_{1}e_{2}$,
$e_{5}=e_{0}(e_{1}e_{2})$ and $e_{6}=e_{0}e_{2}$ respectively.

What ‘allows’ us to treat $e_{0},e_{1},e_{2}$ as of equal importance is the
fact that both these elements and their product are imaginary: taking
$x=e_{0},i=e_{2}$ in (24) reads $e_{0}e_{2}=-e_{2}e_{0}$ while taking
$x=e_{2},i=e_{0}$ reads $e_{2}e_{0}=-e_{0}e_{2}$, which is clearly equivalent.
The same holds for the pairs $(e_{0},e_{1})$ and $(e_{1},e_{2})$ and hence it
follows that when presented with $\mathcal{O}$ there is no meaningful way to
tell whether if it was created as $D_{\gamma}(D_{\beta}(D_{\alpha}(F)))$, as
$D_{\gamma}(D_{\alpha}(D_{\beta}(F)))$, as
$D_{\beta}(D_{\gamma}(D_{\alpha}(F)))$, as
$D_{\beta}(D_{\alpha}(D_{\gamma}(F)))$, as
$D_{\alpha}(D_{\gamma}(D_{\beta}(F)))$ or as
$D_{\alpha}(D_{\beta}(D_{\gamma}(F)))$, even if the three elements
$e_{0},e_{1},e_{2}$ used as $i$ in each doubling step are fixed and known.

######  Lemma 2.37.

Let $e_{3}=e_{0}e_{1}$, $e_{4}=e_{1}e_{2}$, $e_{6}=e_{0}e_{2}$ and
$e_{5}=-e_{3}e_{2}$ then each of $e_{0}\ldots,e_{6}$ is imaginary and their
squares are as given as in Table 2.

###### Proof.

For the proof we return to the viewpoint of $\mathcal{O}$ as
$D_{\gamma}(D_{\beta}(D_{\alpha}(F)))$. $e_{0},e_{1},e_{2}$ are imaginary with
squares $\alpha,\beta,\gamma$ respectively by definition. $e_{3}=e_{0}e_{1}$
appears in the four-dimensional subalgebra $D_{\beta}(D_{\alpha}(F))$. Viewing
$D_{\alpha}(F)$ as $A$ in equation (25) and $e_{0}$ as $x\in A$, $e_{1}$ as
$i$, we find that $e_{3}^{*}=-e_{3}$ which by (15) means that
$e_{3}\in\operatorname{Im}(\mathcal{O})$. For the remaining three products we
use the same reasoning but with $A=D_{\beta}(D_{\alpha}(F))$ and $i=e_{2}$ to
obtain imaginarity of $e_{4},e_{5},e_{6}$.

To prove the claim about squares we note that since we established that for
$i<j\in\\{0,1,2,3\\}$ we have that both $e_{i}$, $e_{j}$ and $e_{i}e_{j}$ are
imaginary we obtain by alternativity and (16) that
$(e_{i}e_{j})^{2}=e_{i}(e_{j}e_{i})e_{j}=e_{i}(-e_{i}e_{j})e_{j}=-e_{i}^{2}e_{j}^{2}$.
Note that the right hand side is just an ordinary product of scalars in $F$. ∎

From the lemma and (16) it also follows that $e_{i}e_{j}=-e_{j}e_{i}$ whenever
$i\neq j$. The usefulness of Table 2 hinges on the following:

######  Lemma 2.38.

The set $\\{1,e_{0},\ldots,e_{6}\\}$ is a basis of $\mathcal{O}$.

###### Proof.

It suffices to show that the set is linearly indepedent and for that it
suffices to show linear independence of the same set but with $e_{5}$ replaced
by $-e_{5}$. Viewing $\mathcal{O}$ as $D_{\gamma}(D_{\beta}(D_{\alpha}(F)))$
and using the shorthand $A=D_{\beta}(D_{\alpha}(F))$ again, we obtain a
vectorspace decomposition $\mathcal{O}=A\oplus Ae_{2}$ as in Definition 2.17.
By construction $\\{1,e_{0},e_{1},e_{3}\\}\subset A$ and
$\\{e_{2}=1e_{2},e_{4}=e_{1}e_{2},-e_{5}=e_{3}e_{2},e_{6}=e_{0}e_{2}\\}\subset
Ae_{2}$. The decomposition $\mathcal{O}=A\oplus Ae_{2}$ reduces linear
independence of the set $\\{e_{0},\ldots e_{6}\\}$ to linear independence of
the two subsets $\\{1,e_{0},e_{1},e_{3}\\}$ and
$\\{e_{2},e_{4},-e_{5},e_{6}\\}$ but we can say something more. Since the
linear map $R_{e_{2}}$ (right multiplication by $e_{2}$) is invertible (its
inverse being $\frac{1}{\gamma}R_{e_{2}}$) we have that the set
$\\{e_{2}=1e_{2},e_{6}=e_{0}e_{2},e_{4}=e_{1}e_{2},-e_{5}=e_{3}e_{2}\\}$ is
linearly independent if and only if the set $\\{1,e_{0},e_{1},e_{3}\\}$ is.
Using the decomposition $A=D_{\alpha}(F)\oplus(D_{\alpha}(F))e_{1}$ we can
then use similar reasoning to show that $\\{1,e_{0},\ldots,e_{6}\\}$ is linear
independent if and only the two element set $\\{1,e_{0}\\}$ is, and this is
the case by definition. ∎

Having identified the vertex set of our Fano plane we move on to recognizing
the lines.

######  Lemma 2.39.

Let $a,b\in\operatorname{Im}(\mathcal{O})$ be two imaginary elements
satisfying $ab\in\operatorname{Im}(\mathcal{O})$ and let $\ell=\\{a,b,ab\\}$.
Then for every two distinct elements $p,q\in\ell$ there is scalar
$\lambda_{p,q}\in F$ such that $pq=\lambda_{p,q}r$ and $qp=-\lambda_{p,q}r$
where $r$ is the third element in the set (‘line’) $\ell$.

###### Proof.

The relation between $pq$ and $qp$ follows from (16) so we focus on
estabishing the $pq=\lambda r$ for some $\lambda\in F$. The case where $p=a$
and $q=b$ or vice versa is trivial, so we may assume that $q=ab$. For $p=a$ we
can use alternativity to move the brackets and conclude $a(ab)=(aa)b=\lambda
b$ with $\lambda=a^{2}\in F$ as desired. For $p=b$ we have
$b(ab)=-b(ba)=-(bb)a=\lambda a$ with $\lambda=-b^{2}\in F$. ∎

The lemma gives a different way of thinking about the Fano-lines. When we view
every tripple $\\{a,b,c\\}\subset\operatorname{Im}(\mathcal{O})$ as a ‘line’
whenever it satisfies $ab=\lambda c$ for some $\lambda\in F^{\times}$ and some
labeling of the elements of the triple as $a$, $b$ and $c$, then the lemma
states that this definition does in fact not depend on the choice of labeling
and that for any pair $\\{p,q\\}$ of imaginary elements we have that all $r$
making $\\{p,q,r\\}$ into a line are scalar multiples of each other. (Where of
course for some choices of $p,q$ no such $r$ exist.)

Our goal is to establish that (1) every two basis elements $e_{i},e_{j}$ lie
on such a line, i.e. have a product that is imaginary, and (2) that the third
point on that line (which by the last lemma is unique up to scalar
multiplication) can be taken to be the basis element indicated by Table 2. For
four of the seven lines this has already been achieved by the above.

For example: our definition of $e_{3}$ as the product $e_{0}e_{1}$ establishes
that the Fano-line through $e_{0}$ and $e_{1}$ passes through $e_{3}$, which
by the lemma means concretely that $e_{0}e_{1}\in Fe_{3}$, $e_{3}e_{0}\in
Fe_{1}$ and $e_{1}e_{3}\in Fe_{0}$. Similarly we obtain the ‘lines’
$\\{e_{1},e_{2},e_{4}\\}$, $\\{e_{2},e_{3},e_{5}\\}$ and
$\\{e_{0},e_{2},e_{6}\\}$. What remains to be done to obtain the full Fano
structure is verifying that the ‘line’ through $e_{3}$ and $e_{4}$ contains
$e_{6}$ (that is: verifying that $e_{3}e_{4}\in F^{\times}e_{6}$) and that the
line through $e_{0}$ and $e_{4}$ as well as the line through $e_{1}$ and
$e_{6}$ both pass through $e_{5}$. This can be done by explicit computation:

$\displaystyle
e_{4}e_{6}=(e_{1}e_{2})(e_{0}e_{2})=(e_{2}e_{1}^{*})(e_{0}e_{2})=\gamma(e_{1}^{*}e_{0})^{*}=\gamma
e_{0}^{*}e_{1}=-\gamma e_{0}e_{1}=-\gamma e_{3}$ $\displaystyle
e_{0}e_{4}=e_{0}(e_{1}e_{2})=e_{0}(e_{2}e_{1}^{*})=e_{2}(e_{0}^{*}e_{1}^{*})=e_{2}(e_{0}e_{1})=e_{2}e_{3}=e_{5}$
$\displaystyle
e_{6}e_{1}=(e_{0}e_{2})e_{1}=(e_{0}e_{1}^{*})e_{2}=e_{2}(e_{0}e_{1}^{*})^{*}=-e_{2}(e_{0}e_{1})^{*}=-e_{2}e_{3}^{*}=e_{2}e_{3}=e_{5}$

Here the first equation uses (24) and (22), both with $e_{2}$ in the role of
$i$, followed by (4) and (15). The second equation uses (24) and (20), both
with $e_{2}$ in the role of $i$, followed by (15). The third equation uses
(21) and (24), both with $e_{2}$ in the role of $i$, followed by (15). Of
course all applications of (15) rely on Lemma 2.37 above.

We have reached the point that for each two elements from
$\\{e_{0},\ldots,e_{6}\\}$ we know the unique Fano line $\ell$ to which they
belong, and for each line $\ell$ we have computed the scalar $\lambda_{a,b}\in
F$ in the equality $ab=\lambda_{ab}c$ for at least one of the six ways to
asign the labels $a,b,c$ to the elements of $\ell$. This means that we can
compute the remaining multiplications of elements on $\ell$ by the rules
$ba=-ab=-\lambda_{ab}c$, $ac=\lambda_{a,b}^{-1}a^{2}b$, $ca=-ac$,
$cb=\lambda_{ab}^{-1}ab^{2}$ and $bc=-cb$. This completes the verification of
Table 2.

Table 2. Multiplication table of the general Cayley-Algebra $D_{\gamma}(D_{\beta}(D_{\alpha}(F)))$ with respect to the basis $e_{0},\ldots,e_{6}$ described in Section 2.6 | $1$ | $e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$
---|---|---|---|---|---|---|---|---
$1$ | $1$ | $e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$
$e_{0}$ | $e_{0}$ | $\alpha$ | $e_{3}$ | $e_{6}$ | $\alpha e_{1}$ | $e_{5}$ | $\alpha e_{4}$ | $\alpha e_{2}$
$e_{1}$ | $e_{1}$ | $-e_{3}$ | $\beta$ | $e_{4}$ | $-\beta e_{0}$ | $\beta e_{2}$ | $-\beta e_{6}$ | $-e_{5}$
$e_{2}$ | $e_{2}$ | $-e_{6}$ | $-e_{4}$ | $\gamma$ | $e_{5}$ | $-\gamma e_{1}$ | $\gamma e_{3}$ | $-\gamma e_{0}$
$e_{3}$ | $e_{3}$ | $-\alpha e_{1}$ | $\beta e_{0}$ | $-e_{5}$ | $-\alpha\beta$ | $-\beta e_{6}$ | $\alpha\beta e_{2}$ | $\alpha e_{4}$
$e_{4}$ | $e_{4}$ | $-e_{5}$ | $-\beta e_{2}$ | $\gamma e_{1}$ | $\beta e_{6}$ | $-\beta\gamma$ | $\beta\gamma e_{0}$ | $-\gamma e_{3}$
$e_{5}$ | $e_{5}$ | $-\alpha e_{4}$ | $\beta e_{6}$ | $-\gamma e_{3}$ | $-\alpha\beta e_{2}$ | $-\beta\gamma e_{0}$ | $\alpha\beta\gamma$ | $\alpha\gamma e_{1}$
$e_{6}$ | $e_{6}$ | $-\alpha e_{2}$ | $e_{5}$ | $\gamma e_{0}$ | $-\alpha e_{4}$ | $\gamma e_{3}$ | $-\alpha\gamma e_{1}$ | $-\alpha\gamma$

###  2.7. Which fields allow Cayley division algebras?

We are interested in the question in the title of this subsection because it
are the Cayley division algebra that give rise, in Section 3, to $q$-Fano
planes. It was pointed out to me by Matthias Wendt [Wen17] that an answer in
terms of Galois cohomology can be found in Serre’s paper [Ser95]: combining
Theorem 9 of that paper with Theorem 5.7 of Merkurjev-Suslin [MS83] as
indicated in Section 8.2 of [Ser95] we find:

######  Theorem 2.40.

The field $F$ of characteristic $\neq 2$ allows Cayley division algebras if
and only if the Galois cohomology group $H^{3}(F,\mathbb{Z}/2\mathbb{Z})$ is
non-trivial.

The definition and other applications of the groups
$H^{i}(F,\mathbb{Z}/2\mathbb{Z})$ are beyond the scope of the current paper.
Instead we will discuss a class of examples where we can understand quite
concretely what is going on.

Since the elements $e_{i}$ in Table 2 are imaginary we have for a generic
element $x=\eta+\xi_{0}e_{0}+\ldots+\xi_{6}e_{6}\in
D_{\gamma}(D_{\beta}(D_{\alpha}(F)))$ that $x^{*}=\eta
1-\xi_{0}e_{0}-\ldots-\xi_{6}e_{6}\in D_{\gamma}(D_{\beta}(D_{\alpha}(F)))$
(equation (15)) and hence, by (8) and Table 2 we find that

(29)
$n(x)=\eta^{2}-\xi_{0}^{2}\alpha-\xi_{1}^{2}\beta-\xi_{2}^{2}\gamma+\xi_{3}^{2}\alpha\beta+\xi_{4}^{2}\beta\gamma+\xi_{6}^{2}\gamma\alpha-\xi_{5}^{2}\alpha\beta\gamma.$

We are interested in finding fields $F$ and elements $\alpha,\beta,\gamma\in
F$ such that $D_{\gamma}(D_{\beta}(D_{\alpha}(F)))$ is a division algebra.
From (29) and Lemma 2.35 we see (with some relabeling) that this is the case
exactly when the equation

(30)
$\lambda_{0}^{2}-\lambda_{1}^{2}\alpha-\lambda_{2}^{2}\beta-\lambda_{3}^{2}\gamma+\lambda_{4}^{2}\alpha\beta+\lambda_{5}^{2}\beta\gamma+\lambda_{6}^{2}\gamma\alpha-\lambda_{7}^{2}\alpha\beta\gamma=0$

has only the trivial solution $\lambda_{0}=\ldots=\lambda_{7}=0$.

From this observation we recognize one class of examples:

######  Corollary 2.41.

Suppose that $F$ contains a subfield $K$ and elements $\alpha,\beta,\gamma$
that are algebraically independent over $K$ such that $F$ is isomorphic to the
function field $F(\alpha,\beta,\gamma)$. Then
$D_{\gamma}(D_{\beta}(D_{\alpha}(F)))$ is a division algebra.

Our proof is adapted from the proof of Bemerkung 5.3.1 in Kristin Stroth’s PhD
thesis [Str13]. Chapter 5 of that thesis further provides an interesting
geometric perspective on Cayley algebras of this type which will not be
discussed here.

###### Proof.

Let $\lambda_{0},\ldots,\lambda_{7}\in K(\alpha,\beta,\gamma)$ be such that
$(\ref{stroth1})$ holds.

Multiplying both sides of the equation with the lowest common multiple of the
denominators of the $\lambda_{i}^{2}$ we may assume that each $\lambda_{i}$ is
an element of the polynomial ring $K[\alpha,\beta,\gamma]$.

Our goal is to show that (30) only has the zero solution over
$K[\alpha,\beta,\gamma]$. We’ll first show that the same holds for the simpler
equation $\nu_{0}^{2}-\nu_{1}^{2}\alpha=0$.

######  Claim 1.

Let $\nu_{0},\nu_{1}\in K[\alpha,\beta,\gamma]$ be such that
$\nu_{0}^{2}-\nu_{1}^{2}\alpha=0$. Then $\nu_{0}=\nu_{1}=0$.

###### Proof.

Rewriting the equation as $\nu_{0}^{2}=\nu_{1}^{2}\alpha$ and interpreting
both sides as polynomials in $\alpha$ with coefficients in $K[\beta,\gamma]$
we see that the left hand side has even degree, while the right hand side has
odd degree (adopting the convention that the zero polynomial has every
degree). Hence both sides must be zero. ∎

We want to use the same reasoning to show that the ‘intermediate level’
equation $\mu_{0}-\mu_{1}\alpha-\mu_{2}\beta+\mu_{3}\alpha\beta=0$ also only
has the zero solution, but we need an extra step:

######  Claim 2.

Let $\mu_{0},\mu_{1}\in K[\alpha,\beta,\gamma]$. Then viewed as a polynomial
in $\beta$, $\xi\coloneqq\mu_{0}^{2}-\mu_{1}^{2}\alpha$ has even degree.

###### Proof.

Let $m_{0},m_{1}$ be the degrees of $\mu_{0},\mu_{1}$ respectively when viewed
as polynomials in $\beta$ and let $m=\max(m_{0},m_{1})$. When $m_{0}\neq
m_{1}$ we have that the degree of $\xi=\mu_{0}^{2}-\mu_{1}^{2}\alpha$ equals
$2m$ and we are done. When $m_{0}=m_{1}=m$ define $\nu_{0},\nu_{1}$ as the
leading (i.e. degree $m$) terms of $\mu_{0},\mu_{1}$ respectively when viewed
as a polynomial in $\beta$. The degree $2m$ term in $\xi$ equals
$\nu_{1}^{2}-\nu_{2}^{2}\alpha$. Since $\xi$ clearly contains no terms of
degree higher than $2m$ we see that the only scenario in which the degree of
$\xi$ does not equal $2m$ is when its degree $2m$ term
$\nu_{0}^{2}-\nu_{1}^{2}\alpha$ equals zero. But by Claim 1 this only happens
when $\nu_{0}=\nu_{1}=0$, which, as the $\nu_{i}$ were defined as the leading
terms of the $\mu_{i}$ implies that the $\mu_{i}$ are zero. Hence we see that
either the degree of $\xi$ is $2m$ or $\xi=0$, in which case we consider the
degree of $\xi$ to be even as well. ∎

With this result under our belt we can lift the proof of Claim 1 one level
higher:

######  Claim 3.

Let $\mu_{0},\mu_{1},\mu_{2},\mu_{3}\in K[\alpha,\beta,\gamma]$ be such that
$\mu_{0}-\mu_{1}\alpha-\mu_{2}\beta+\mu_{3}\alpha\beta=0$. Then
$\mu_{0}=\mu_{1}=\mu_{2}=\mu_{3}=0$.

###### Proof.

We can rewrite the equation as

$(\mu_{0}^{2}-\mu_{1}^{2}\alpha)=(\mu_{2}^{2}-\mu_{3}^{2}\alpha)\beta.$

From Claim 2 we see that the left hand side has even degree when viewed as a
polynomial in $\beta$ while the right hand side has odd degree. It follows
that both sides of the equations equal zero, so that
$\mu_{0}^{2}-\mu_{1}^{2}\alpha=0$ and $\mu_{2}^{2}-\mu_{3}^{2}\alpha=0$. But
Claim 1 then implies that $\mu_{0}=\mu_{1}=0$ and that $\mu_{2}=\mu_{3}=0$. ∎

From this point on it is clear how we will proceed. Similar to how we derived
Claim 2 from Claim 1 we derive from Claim 3 that expressions of the form
$\mu_{0}^{2}-\mu_{1}^{2}\alpha-\mu_{2}^{2}\beta+\mu_{3}^{2}\alpha\beta$ (with
$\mu_{0},\mu_{1},\mu_{2},\mu_{3}\in K[\alpha,\beta,\gamma]$) have even degree
when viewed as polynomials in $\gamma$.

We then use this to conclude that the left and right hand sides of the
following equivalent reformulation of (30) have different degrees as
polynomials in $\gamma$ and hence are both zero:

(31)
$(\lambda_{0}^{2}-\lambda_{1}^{2}\alpha-\lambda_{2}^{2}\beta+\lambda_{4}^{2}\alpha\beta)=(\lambda_{3}^{2}-\lambda_{5}^{2}\beta-\lambda_{6}^{2}\alpha+\lambda_{7}^{2}\alpha\beta)\gamma.$

In summary we find that

(32)
$\displaystyle\lambda_{0}^{2}-\lambda_{1}^{2}\alpha-\lambda_{2}^{2}\beta+\lambda_{4}^{2}\alpha\beta$
$\displaystyle=0$ (33)
$\displaystyle\lambda_{3}^{2}-\lambda_{5}^{2}\beta-\lambda_{6}^{2}\alpha+\lambda_{7}^{2}\alpha\beta$
$\displaystyle=0$

and Claim 3 then tells us that
$\lambda_{0}=\lambda_{1}=\lambda_{2}=\lambda_{4}=0$ and that
$\lambda_{3}=\lambda_{5}=\lambda_{6}=\lambda_{7}=0$, as we wanted to show. ∎

######  Remark 2.42.

The proof that over the fields $K(\alpha,\beta,\gamma)$ equation (30) has no
non-zero solutions relies on the _multiplicative_ properties of squares,
notably that the product of two squares is a square and the product of a
square and a non-square is a non-square. At the other extreme we find the
subfields of $\mathbb{R}$ where we can get to the same conclusion by looking
at the _additive_ properties of squares over such field. Since in a subfield
of $\mathbb{R}$ a sum of squares is always non-negative, we find that over
such field equation (30) has no non-zero solutions whenever
$\alpha<0,\beta<0,\gamma<0$.

It is tempting to see if we can construct further examples that use some
mixture of additive and multiplicative properties e.g. by somewhat loosening
the condition that $\alpha,\beta,\gamma$ are algebraically independent over
$K$ in Corollary 2.41 while being careful about what it means to be square in
the resulting field. We won’t pursue that direction here, however.

###  2.8. Some structure theory

So far we have focussed on the similarities rather than differences between
split and divsion Cayley-Dickson algebras. However, since the method explored
in Section 3 that constructs $q$-Fano planes from Cayley division algebras
fails to do so in the split case (for reasons explored in depth in Section 4)
there must also be considerable differences between the two cases. Informally
speaking the most important difference (to us) is that, unlike their division
algebra counterparts, the split Cayley-Dickson algebras contain _subalgebras_
that are quite different from the Cayley-Dickson algebras of either type. The
prototypical example of such a subalgebra is the algebra $U_{3}$ of upper
triangular 2-by-2-matrices sitting inside the split quaternion algebra $M_{4}$
of all two-by-two matrices. Being three-dimensional this algebra cannot be a
Cayley-Dickson algebra, even if it is closed under the star operator defined
in Example 2.31 and hence is a quadratic algebra with a strong involution. In
this subsection we recall some notions from the structure theory of
alternative algebras that enable us to articulate the differences between this
algebra and the Cayley-Dickson algebras.

######  Definition 2.43.

A two-sided _ideal_ in an alternative algebra $A$ is a linear subspace $I$
such that $ai\in I$ and $ia\in I$ for every $a\in A,i\in I$. An algebra $A$ is
called _simple_ if its only two sided ideals are $\\{0\\}$ and $A$ itself.

The definition of ideal ensures that when $I$ is an ideal in $A$ then the
multiplication in $A$ descends to a well-defined multiplication on the
quotient vector space $A/I$. When $A$ is alternative or even associative then
so is the quotient algebra $A/I$. It is easy to see that all division algebras
are simple. Perhaps more surprising is that the split 4- and 8-dimensional
Cayley-Dickson algebras are simple as well. (We’ll come back to that at the
end of this section). The algebra $U_{3}$ of upper triangular matrices is
_not_ simple as its subspace consisting of strict upper triangular matrices is
an ideal. The elements of this ideal also serve as an example to the following
concept, that will be important in the sequel:

######  Definition 2.44.

An element $x$ of an alternative algebra $A$ is called _nilpotent_ if there
exists a natural number $n$ such that $x^{n}=0$.

######  Lemma 2.45.

Let $x\neq 0$ be an element of an alternative Cayley-Dickson algebra $O$. Then
the following are equivalent:

  1. (1)

$x$ is nilpotent

  2. (2)

$x^{2}=0$

  3. (3)

$x$ is a zero-divisor and $x\in\operatorname{Im}(O)$

###### Proof.

$(1)\Rightarrow(2).$ Let $x$ be nilpotent and let $k$ be the smallest number
such that $x^{k}=0$. Since clearly $0$ is the only element for which $k=1$,
and clearly $0^{2}=0$ we will assume in the sequel that $k\geq 2$. From
multiplicativity of the function $n$ (equation 17) we find that $n(x)^{k}=0\in
F$ and hence $n(x)=0$. Multiplying both sides of (8) by $x^{k-2}$ we find that
$2\tau(x)x^{k-1}=0$. By minimality of $k$ this implies that $\tau(x)=0$.
Plugging this back into the original equation (8) we find that $x^{2}=0$.

$(2)\Rightarrow(3).$ Assume $x^{2}=0$. It is clear that $x$ is a divisor of
zero. The fact that $x\in\operatorname{Im}(O)$ follows from (10).

$(3)\Rightarrow(1)$ Let $x$ be an imaginary divisor of zero. By the latter
property there exist a $y$ such that $xy=0$ or $yx=0$. In the latter case we
have that $0^{*}=(yx)^{*}=x^{*}y^{*}=-xy^{*}=x(-y^{*})$, so after replacing
$y$ with $-y^{*}$ if needed we see that there exist non-zero $y$ such that
$xy=0$. Now by alternativity we can move the brackets in the following
expression to obtain $0=x(xy)=(x^{2})y$. By $(\ref{im})$, $x^{2}\in F1$ so the
right hand side is a scalar multiple of the non-zero element $y$ and hence we
find that $x^{2}=0$ proving (2) and hence (1). ∎

######  Definition 2.46.

An element $x$ in an alternative algebra $A$ is called _strongly nilpotent_ if
$xy$ is nilpotent for all $y\in A$ or, equivalently, if $yx$ is nilpotent for
all $y\in A$. The set of all strongly nilpotent elements is called the
_Jacobson radical_ of $A$ and will be denoted $J(A)$.

When $A$ is associative, it is clear that $J(A)$ is a two-sided ideal in $A$.
In fact this is already true, though highly non-obviously so, when $A$ is
merely alternative. A proof of this fact (originally due to Zorn [Zor41]) can
be found in Chapter 3 of Schafer’s book [Sch66] (though unfortunately only in
the 1966 edition and not in the 1961 edition that is freely available online),
as can the proof of the following two theorems:

######  Theorem 2.47.

Let $A$ be an alternative algebra. The following are equivalent:

  1. (1)

$J(A)=\\{0\\}$

  2. (2)

$A$ is isomorphic as an algebra to a direct sum of simple alternative
algebras.

The algebra direct sum $A_{1}\oplus\ldots\oplus A_{n}$ of algebras $A_{i}$ is
the vector space direct sum $A_{1}\oplus\ldots\oplus A_{n}$ equiped with the
_pointwise_ multiplication. In other words: multiplication of two elements in
the same summand $A_{i}$ is just the ordinary multiplication in the algebra
$A_{i}$ while the multiplication of elements from different summands equals
zero. This multiplication defined on elements of the summands is then extended
linearly to all of $A$. It is clear that this construction preserves
alternativity and associativity.

######  Definition 2.48.

An algebra satisfying the equivalent conditions of Theorem 2.47 is called
_semi-simple_.

An example of a semi-simple algebra which is not simple is the two-dimensional
split Cayley Dickson algebra which we met in Proposition 2.30. This algebra is
isomorphic to the algebra $D_{2}$ of 2-by-2 diagonal matrices which sits as a
subalgebra inside the algebra $U_{3}$ introduced at the beginning of this
section. Since it is not hard to show that $J(U_{3})$ equals the ideal of
strict upper triangular matrices we find that $U_{3}$ decomposes (as a vector
space, not as an algebra) as a sum of subalgebras $U_{3}=D_{2}\oplus
J(U_{3})$. It turns out that this is part of a more general pattern:

######  Theorem 2.49.

Let $A$ be an alternative algebra with Jacobson radical $J(A)$. Then $A$
contains a semi-simple subalgebra $S$ such that as a vectorspace $A=S\oplus
J(A)$ and such that the resulting vectorspace isomorphism $S\cong A/J(A)$ is
in fact an isomorphism of algebras.

The fact $A/J(A)$ even _is_ an algebra follows from Zorn’s result quoted
above. The fact that the resulting quotient algebra is semi-simple is then a
straightforward consequence of the definitions. The really surprising part is
that $A$ already had this semi-simple quotient within it all along.

######  Notation 2.50.

The subalgebra $S$ of Theorem 2.49 will be called the _semi-simple part_ of
$A$.

In the cases we are interested in we can give an alternative characterisation
of the Jacobson radical:

######  Lemma 2.51.

Let $A$ be an alternative algebra with a strong involution (e. g. any
subalgebra of a Cayley algebra) and let $x\in A$. Then $x$ is strongly
nilpotent if and only if it is _strongly imaginary_ , that is: if and only if
$xy\in\operatorname{Im}(A)$ for every $y\in A$. (And, equivalently,
$yx\in\operatorname{Im}(A)$ for every $y\in A$.)

###### Proof.

The ‘only if’ direction is immediate from implication $1\Rightarrow 3$ of
Lemma 2.45. For the ‘if’ direction let $x$ be strongly imaginary. Let $y\in
A$. We first prove the parenthetical statement that
$yx\in\operatorname{Im}(A)$. Since $x1\in\operatorname{Im}(A)$, we see that
strongly imaginary elements are imaginary themselves. Applying (15) to the
imaginary elements $x$ and $xy^{*}$ we find that $(yx)^{*}=-yx$:
$(yx)^{*}=x^{*}y^{*}=-xy^{*}$ $=(xy^{*})^{*}=yx^{*}=-yx$. This implies that
$yx$ is imaginary by (15).

Next we establish that $x$ itself is nilpotent. Since $x$ is strongly
imaginary, $x^{2}\in\operatorname{Im}(A)$. But since $x$ is ‘ordinarily’
imaginary, $x^{2}\in F1$. It follows that $x^{2}\in
F1\cap\operatorname{Im}(A)=\\{0\\}$ and hence $x^{2}=0$.

Finally to see that $xy$ is nilpotent as well, we compute
$(xy)^{2}=-(xy)^{*}(xy)=-y^{*}x^{*}xy=y^{*}xxy=y^{*}0y=0$. ∎

Using this lemma it is easy to show inductively:

######  Theorem 2.52.

Let $A$ be a Cayley-Dickson algebra of dimension 1, 2, 4 or 8, then $A$ is
semi-simple.

We already saw that the same is not true for all _subalgebras_ of the split
quaternion and Cayley algebras.

We conclude this section by proving of the following variant of Theorem 2.52:

######  Theorem 2.53.

Let $A$ be a Cayley-Dickson algebra of dimension 1, 4 or 8, then $A$ is
simple.

######  Definition 2.54.

The _center_ of an algebra $A$ is the vector space of all elements $c\in A$
that commute and associate with every element of $A$. Clearly $F1$ is
contained in the center of any algebra $A$. When $F1$ equals the entire center
of $A$ then $A$ is called _central_.

Since the center of a direct sum of algebras contains the direct sum of the
centers of the summands, it follows that a semi-simple algebra can only be
central when it consists of a single summand and thus is simple. We conclude
that Theorem 2.53 follows from Theorem 2.52 together with the following
proposition:

######  Proposition 2.55.

All Cayley-Dickson algebras (not necessarily alternative) of dimension $\neq
2$ are central.

###### Proof.

For the Cayley-Dickson algebra $F$ the statement is clearly true so we can
restict our attention to algebras of the form $D_{\gamma}(A)$, the Dickson
double of a Cayley-Dickson algebra $A$. Let $c$ be in the center of
$D_{\gamma}(A)$. We can write $c=a+ib$ with $a,b\in A$ and $i$ as in
$(\ref{pq1}-\ref{xistar})$. Expanding both sides of the equality $ci=ic$ into
the standard form $r+is$ using $(\ref{pq1}-\ref{pq3})$ and comparing terms
along the vectorspace decomposition $D_{\gamma}(A)=A\oplus iA$, we find that
$a=a^{*}$ and $b=b^{*}$. It follows by Corollary 2.19 that $a\in F1$ and $b\in
F1$.

Since $c$ and $a$ are both in the center of $D_{\gamma}(A)$ so is $c-a=ib$. We
distinguish two cases: $b=0$ and $b\neq 0$. First suppose that $b\neq 0$.
Since $b\in F1$ we conclude that $b$ is invertible and that, since $ib$ is in
the center of $D_{\gamma}(A)$, so is $i$. This means in particular that for
any $x\in A$ we have that $x=x^{*}$ by (24). By Corollary 2.19 this implies
that $A=F1$ and hence that $D_{\gamma}(A)$ is two-dimensional.

Conversely when $D_{\gamma}(A)$ is not two-dimensional we conclude that $b=0$.
But this implies that $c=a\in F1$. Since $c$ was an arbitrary element of the
center of $D_{\gamma}(A)$, we conclude that $D_{\gamma}(A)$ is central. ∎

######  Remark 2.56.

It is known that the dimension of any central simple _associative_ algebra is
a square. Hence the smallest examples beyond $F$ are 4-dimensional and in the
literature on associative algebras these are called _quaternion algebras_. We
just proved that a quaternion algebra in our sense, i.e. a 4-dimensional
Cayley-Dickson algebra, is indeed central and simple. The converse also holds:
any 4-dimensional central simple algebra $Q$ contains, as a subalgebra, a
quadratic field extension $C$ of $F$ such that $Q$ is isomorphic to a Dickson
double of $C$. (The proof of this latter fact can be found in any text on
associative algebras, e.g. [Pie12], [Vig80].) Thus the two notions of
quaternion algebra are equivalent.

##  3\. $q$-Fano planes from Cayley Division algebras

In this section we will deduce the existence of a $2$-$(7,3,1)_{F}$-subspace
design from the existence of a Cayley division algebra over $F$. As seen in
Section 2.5 this latter existence holds only for certain infinite non-
algebraically closed fields. The first three results of this section (Lemmas
3.1 and 3.2 and Proposition 3.3) do hold over general fields of characteristic
unequal to 2.

######  Lemma 3.1.

Let $H$ be a unital subalgebra of a Cayley algebra $O$ over $F$. Then $H$ is
closed under $*$ and $\operatorname{Im}(H)=H\cap\operatorname{Im}(O)$.

###### Proof.

$O$ is a quadratic algebra by Corollary 2.19 and Theorem 2.15. It is immediate
from the definition of quadratic algebras that subalgebras of quadratic
algebras are quadratic again, so that $\operatorname{Im}(H)$ is well defined.
The statement that $\operatorname{Im}(H)=H\cap\operatorname{Im}(O)$ is
immediate from characterization (10) of the former space. Let $x\in H$. As in
Section 2.3, we write $\tau$ and $\operatorname{Im}$ respectively for the
projection operators along the decomposition $O=F1\oplus\operatorname{Im}(O)$.
In particular for $x\in H$ there exist $\tau(x)\in
F1,\operatorname{Im}(x)\in\operatorname{Im}(O)$ such that
$x=\tau(x)+\operatorname{Im}(x)$. Since $1\in H$ by assumption and $x\in H$ by
definition we have that $\operatorname{Im}(x)=x-\tau(x)\in H$ and hence that
$x^{*}=\tau(x)-\operatorname{Im}(x)\in H$. ∎

######  Lemma 3.2.

Let $u,v$ be linearly independent imaginary elements of a Cayley algebra $O$
over $F$. Then the algebra $\langle u,v\rangle$ generated by $u$ and $v$ is
equal as a vector space to the space $\operatorname{span}(\\{1,u,v,uv\\})$ and
hence has dimension either 3 or 4.

###### Proof.

It is clear that $\operatorname{span}(1,u,v,uv)$ is contained in $\langle
u,v\rangle$, which in turn is associative by alternativity of $O$. To get the
converse inclusion it suffices to show that the linear space
$\operatorname{span}(1,u,v,uv)$ is closed under the multiplication. By
associativity and the fact that $u^{2},v^{2}\in F1$ by (10), we don’t have to
worry about the products $u^{2},v^{2},u^{2}v$ and $uv^{2}$ and the definition
of quadratic algebra tells us that $(uv)^{2}\in\operatorname{span}(1,uv)$.
Thus it only remains to show that the products $vu$, $uvu$ and $vuv$ are all
in $\operatorname{span}(1,u,v,uv)$.

By (4) and (15) we find that $(uv)^{*}=v^{*}u^{*}=vu$ and hence
$(uv+vu)^{*}=(uv+vu)$. The fact that $*$ is a strong involution by Corollary
2.19 then implies that $uv+vu\in F1$ and hence that
$vu\in\operatorname{span}(1,uv)\subseteq\operatorname{span}(1,u,v,uv)$. It
follows that
$uvu\in\operatorname{span}(u,u^{2}v)=\operatorname{span}(u,v)\subset\operatorname{span}(1,u,v,uv)$
and
$vuv\in\operatorname{span}(v,uv^{2})=\operatorname{span}(v,u)\subset\operatorname{span}(1,u,v,uv)$
and we conclude that the latter space is closed under multiplication.

This proves that $\langle u,v\rangle=\operatorname{span}(1,u,v,uv)$. ∎

######  Proposition 3.3.

Let $O$ be a Cayley algebra over $F$ and let $U\subseteq\operatorname{Im}(O)$
be a 2-dimensional subspace not containing any nilpotent elements. Then
$\langle U\rangle$ is a four-dimensional associative subalgebra of $O$.

###### Proof.

Let $\\{u,v\\}$ be a basis of $U$. From Lemma 2.6 we know that $\langle
U\rangle=\langle u,v\rangle$. The latter subalgebra is associative by
alternativity of $O$ and from Lemma 3.2 we know that it is either 3 or
4-dimensional. What remains to be shown is that this space is 4- rather than
3-dimensional. We will do this by showing that $\langle u,v\rangle$ is
isomorphic to the Dickson-Double of the algebra $\langle u\rangle$, which
itself is two-dimensional since $O$ is quadratic (Definition 2.11). This is
will be the proof announced in Remark 2.24.

Since $u$ is not nilpotent, we see that $n(u)$, which equals $-u^{2}$ by (8),
is non-zero.

Let $i=v+\frac{\tau(uv)}{n(u)}u$. Since $\frac{\tau(uv)}{n(u)}$ is just a
scalar in $F$ this is an element of the linear space
$\operatorname{span}(u,v)\subseteq\langle u,v\rangle\cap\operatorname{Im}(O)$.
In particular we have by (15) that

(34) $i^{*}=-i$

By construction we also have that $\tau(ui)=0$, where we exploited that, since
$u\in\operatorname{Im}(O)$, we have that $u^{2}=-n(u)1$ by (8). It follows
that $ui\in\operatorname{Im}(O)$ and by bilinearity of the product we find
that $xi\in\operatorname{Im}(O)$ for every $x\in\langle
u\rangle=\operatorname{span}(1,u)$. We conclude from (15), (4) and (34)
respectively that

(35) $xi=-(xi)^{*}=-(i^{*})x^{*}=ix^{*}$

from which it follows that the pair $A=\langle
u\rangle,i=v+\frac{\tau(uv)}{n(u)}u$ satisfies equation (24). Remark 2.20 then
implies that it satisfies (20, 21, 22).

Define $\gamma\in F$ by $i^{2}=\gamma 1$. Since, by the assumptions of the
proposition, $i$ is not nilpotent, $\gamma\neq 0$. Now the fact that the pair
$A=\langle u\rangle,i=v+\frac{\tau(uv)}{n(u)}u$ satisfies (20, 21, 22) implies
by Lemma 2.18 that there is a non-zero homomorphism from the ‘abstract’ four-
dimensional Cayley-Dickson algebra $D_{\gamma}(\langle u\rangle)$ to
$B\coloneqq\langle u,v\rangle$.

Since the former of these algebras is simple by Theorem 2.53 and the kernel of
any homomorphism is a two-sided ideal, this homomorphism is injective and
hence $\dim(B)\geq 4$. Since we already established that $\dim(\langle
u,v\rangle)\leq 4$ in Lemma 3.2 we find that in fact $\dim(\langle
u,v\rangle)=4$ and $\langle u,v\rangle\cong D_{\gamma}(\langle u\rangle)$. ∎

######  Theorem 3.4 (Existence of $q$-Fano plane).

Let $F$ be a field of characteristic unequal to 2 over which there exist at
least one Cayley division algebra $O$. Then there exists a
$2$-$(7,3,1)_{F}$-subspace design over $F$. More precisely: let $O$ be a
Cayley division algebra over $F$, let $V=\operatorname{Im}(O)$ and let

$\mathfrak{B}=\\{\operatorname{Im}(H)\colon H\textnormal{ is a 4-dimensional
associative subalgebra of }O\\}.$

Then $\dim V=7$, $\dim(B)=3$ for every $B\in\mathfrak{B}$ and every two-
dimensional subspace $U\subseteq V$ is contained in a unique element $B$ of
$\mathfrak{B}$.

###### Proof.

Let $O,V,\mathfrak{B}$ be as in the statement of the theorem. The given
dimensions of $V$ and $B$ for $B\in\mathfrak{B}$ follow from Lemma 2.16. Let
$U\subseteq V$ be 2-dimensional. Write $H=\langle U\rangle$. Since $O$
contains no divisors of zero, it certainly contains no nilpotent elements. By
Proposition 3.3 then, $H$ is a 4-dimensional associative subalgebra of $O$
containing $U$.

_Existence of $B\in\mathfrak{B}$ containing $U$._ By construction, $H$
contains $U$. Also by definition $U\subseteq V$ so it follows that $U\subseteq
H\cap V$. But by Proposition 3.1 this latter space equals
$\operatorname{Im}(H)\in\mathfrak{B}$.

_Uniqueness of $B\in\mathfrak{B}$ containing $U$._ Suppose there are
$B_{1},B_{2}\in\mathfrak{B}$ such that $U\subseteq B_{1}\cap B_{2}$. By
definition there exist four-dimensional associative algebras
$H_{1},H_{2}\subseteq O$ such that $\operatorname{Im}(H_{1})=B_{1}$,
$\operatorname{Im}(H_{2})=B_{2}$. Since both $H_{1}$ and $H_{2}$ contain $U$
and both are subalgebras, they both contain $\langle U\rangle$. But since
$\dim H_{1}=\dim\langle U\rangle$ by Proposition 3.3 we have that
$H_{1}=\langle U\rangle$ and similarly $H_{2}=\langle U\rangle$. It follows
that $H_{1}=H_{2}$ and hence
$B_{1}=\operatorname{Im}(H_{1})=\operatorname{Im}(H_{2})=B_{2}$. ∎

##  4\. The subalgebras generated by two-dimensional subspaces of
$\operatorname{Im}(O)$

Proposition 3.3 showed that if the two-dimensional subspace
$U\subset\operatorname{Im}(O)$ contains no nilpotent elements, then $\langle
U\rangle$ is a four-dimensional simple associative non-commutative algebra. In
a sense, the demand that $U$ contains no nilpotents ‘feels’ a bit too strong
(though weak enough to cover all cases in case $O$ is a division algebra): the
only way it is used in the proof is to guarantee that we can pick a non-
nilpotent basis element $u$ of $U$ and that the element $i$ constructed from
$u$ and the other, arbitrary, basis element $v$ is not a nilpotent. In the
current section we will see that in the case that $O$ is split (and nilpotent
elements hence do exist) the class of two-dimensional subspaces generating a
four-dimenional simple associative non-commutative subalgebra is indeed
strictly larger than the class of two-dimensional subspaces without
nilpotents, and that moreover some other two-dimensional subspaces of
$\operatorname{Im}(O)$ (containing nilpotents) generate non-simple associative
algebras that are nevertheless non-commutative and four-dimensional. For the
purpose of obtaining a $q$-Fano plane by analogy to Theorem 3.4 these non-
simple algebras are clearly ‘good enough’. What we really want to know is
whether every algebra $\langle U\rangle$ is necessarily four-dimensional.
Unfortunately, when $O$ is split, the answer is no. Before giving an example
of a space $U$ generating a three-dimensional subalgebra we discuss some other
properties of these subspaces so that we know where to look for them.

###  4.1. Subspaces generating three-dimensional subalgebras

We begin with a trivial but crucial observation:

######  Lemma 4.1.

Let $u,v\in\operatorname{Im}(O)$. Then $\dim\langle u,v\rangle=3$ if and only
if $uv\in\operatorname{span}(1,u,v)$. In terms of vector spaces rather than
elements: the algebra $\langle U\rangle$ generated by a two-dimensional
subspace $U\subset\operatorname{Im}(O)$ is 3-dimensional if and only if
$\operatorname{Im}(\langle U\rangle)=U$ and $\langle U\rangle=F1\oplus U$ as a
vectorspace. In other words $\dim\langle U\rangle=3$ if and only if the
vectorspace $F1\oplus U$ is closed under multiplication.

This gives us _some_ algortithm of probing $O$ for such subspaces: just pick
random pairs of elements and see where their product lands. However we will
try and be more efficient by understanding in more detail the structure of
such subspaces and the algebras they generate.

Two-dimensional subspaces of $\operatorname{Im}(O)$ that generate only a
three-dimensional subalgebra come in two flavours which we will call _type Z_
and _type U_ :

######  Proposition 4.2.

Let $U\subset\operatorname{Im}(O)$ be such that $\dim U=2$, $\dim\langle
U\rangle=3$. Then either:

  * •

$uv=0$ for every $u,v\in U$ – we will call such spaces _type Z_ spaces because
the multiplication on the space $U$ acts as the zero map,

or:

  * •

$U$ contains a single line $Fu$ of nilpotent elements and for every $v\in U$
not on that line we have that $v^{2}\neq 0$ but $uv\in Fu$ (and hence
$vu=-uv\in Fu)$. We will call spaces of the latter type to be of _type U_ for
reasons explained later.

Conversely it is easy to see that when $U$ is of type either $U$ or $Z$ then
$F1\oplus U$ is closed under multiplication and hence equal to the algebra
$\langle U\rangle$.

###### Proof.

Let $U$ be such that $\langle U\rangle=F1\oplus U$ and let $\\{u,v\\}$ be a
basis of $U$. By Proposition 3.3 we may assume without loss of generality that
$u$ is nilpotent and hence (by Lemma 2.45) that $u^{2}=0$. We know that
$v^{2}=\alpha 1$ and $uv=\beta 1+\gamma u+\delta v$ for some
$\alpha,\beta,\gamma,\delta\in F$. From $u(uv)=(uu)v$ we find that $\beta
u+0+\delta\beta 1+\delta\gamma u+\delta^{2}v=0$ so that $\beta=\delta=0$ and
hence that $uv=\gamma u$. From $(uv)v=u(v^{2})$ we then find that
$\gamma^{2}u=\alpha u$ so that $\gamma^{2}=\alpha$.

Now we distinguish two cases: $\gamma=0$ and $\gamma\neq 0$. When $\gamma=0$
we have that $v^{2}=\alpha=\gamma^{2}=0$ and from $\beta=\gamma=\delta=0$ we
have that $uv=0$. We recall that moreover we had $u^{2}=0$ from the start. It
follows easily that the product of any two elements in
$U=\operatorname{span}(u,v)$ is zero and hence that $U$ is of type Z.

When $\gamma\neq 0$ we find from $\beta=\delta=0$ that $uv=\gamma u\in Fu$ and
hence in particular $uv\in\operatorname{Im}(O)$ so that
$vu=v^{*}u^{*}=(uv)^{*}=-uv=-\gamma u\in Fu$ as well. Moreover, from
$\alpha=\gamma^{2}$ we see that $v^{2}\neq 0$. In particular if $x=\epsilon
u+\zeta v$ with $\zeta\neq 0$ is a generic element of $U$ not on $Fu$ we find
that $ux=\zeta\gamma u\in Fu$ but $x^{2}=\zeta^{2}\gamma^{2}\neq 0\in F$ so
that $U$ is of type U. ∎

The algebra structure of $\langle U\rangle$ in case $U$ is of type $Z$ is very
easy to understand. Every element $\langle U\rangle$ is of the form $\alpha
1+u$ with $\alpha\in F,u\in U$ and multiplication is given by $(\alpha
1+u)(\beta 1+v)=\alpha\beta 1+(\beta u+\alpha v)$.

In case of spaces $U$ of type U we have the following result, explaining the
name:

######  Proposition 4.3.

Let $U$ be a two-dimensional type U space. Then the three-dimensional algebra
$\langle U\rangle$ is isomorphic to the algebra of upper triangular 2-by-2
matrices. More strongly: for every basis $u,v$ of $U$ with
$u^{2}=0,v^{2}=\gamma^{2}1$ as in the proof of Proposition 4.2 there is a
unique isomorphism $\phi_{u,v}\colon\langle
U\rangle\to\left\\{\begin{pmatrix}\zeta&\eta\\\
0&\theta\end{pmatrix}\colon\zeta,\eta,\theta\in
F\right\\}\subset\operatorname{Mat}(2,F)$ given by
$\phi(1)=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$,
$\phi(u)=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$,
$\phi(v)=\begin{pmatrix}\gamma&0\\\ 0&-\gamma\end{pmatrix}$.

Moreover if we define the involution $*$ on $\operatorname{Mat}(2,F)$ as in
Example 2.31 then each of the maps $\phi_{u,v}$ preserves the $*$-structure.
In particular the space $U=\operatorname{Im}(\langle U\rangle)$ is mapped to
the space of traceless matrices, the Jacobson radical $J(\langle U\rangle)=Fu$
is mapped to the space of strictly upper triangular matrices and the
subalgebra $\operatorname{span}(1,v)$, which is isomorphic to the (necessarily
split) Cayley-Dickson algebra $D_{\gamma^{2}}(F)$ is mapped to the subalgebra
of diagonal matrices in $\operatorname{Mat}(2,F)$.

###### Proof.

Clearly each of the $\phi_{u,v}$ is a linear isomorphism. It suffices to show
that it preserves the multiplication and the involution, but this is clear
from the properties of the basis $u,v$ derived in the proof of Proposition
4.2. ∎

###  4.2. Multiplication table of the split Cayley-Algebra

Now that we understand the structure of the two-dimensional subspaces of
$\operatorname{Im}(O)$ generating three-dimensional subalgebras and that of
the algebras they generate, we return to the question of their existence. By
Proposition 3.3 such subspaces can only exist when $O$ contains zero-divisors,
which by Theorem 2.28 means that it is unique up to isomorphism. This has the
advantage that we can give a very explicit description of such Cayley algebras
$O$. We will do so using Table 3 taken from [CRE16]. The authors’ remarkable
choice of not using 1 as a basis element has the advantage that it highlights
various unexpected symmetries among the generators.

######  Lemma 4.4.

The split Cayley algebra over $F$ has a basis consisting of two idempotent
elements $p_{1}$, $p_{2}$ satisfying $p_{1}+p_{2}=1$ and six nilpotent
elements $q_{1},q_{2},q_{3},r_{1},r_{2},r_{3}$ with multiplication as given by
Table 3. The involution acts by $p_{1}^{*}=p_{2}$, $p_{2}^{*}=p_{1}$ and
$q_{i}^{*}=-q_{i}$, $r_{i}^{*}=-r_{i}$ for $i=1,2,3$.

###### Proof.

By Thm. 2.28 and Prop. 2.31 we can realize the split Cayley algebra over $F$
as $D_{-1}(\operatorname{Mat}(2,F))$, which in particular means that as a
vector space it decomposes as $\operatorname{Mat}(2,F)\oplus
i\operatorname{Mat}(2,F)$ as in Definition 2.17. We pick
$p_{1},p_{2},q_{1},r_{1}$ in the first summand as follows:

$p_{1}=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix};\quad
p_{2}=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix};\quad q_{1}=\begin{pmatrix}0&1\\\
0&0\end{pmatrix};\quad r_{1}=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}.$

and then define $q_{2},r_{2},q_{3},r_{3}$ in the second summand by

$q_{2}=ip_{2};\quad r_{2}=ip_{1};\quad q_{3}=-ir_{1};\quad r_{3}=-iq_{1}.$

Then the correctness of Table 3 follows from (20, 21, 22). ∎

Table 3. Multiplication table of the split Cayley algebra in the basis described by Lemma 4.4 | $p_{1}$ | $p_{2}$ | $q_{1}$ | $q_{2}$ | $q_{3}$ | $r_{1}$ | $r_{2}$ | $r_{3}$
---|---|---|---|---|---|---|---|---
$p_{1}$ | $p_{1}$ | $0$ | $q_{1}$ | $q_{2}$ | $q_{3}$ | $0$ | $0$ | $0$
$p_{2}$ | $0$ | $p_{2}$ | $0$ | $0$ | $0$ | $r_{1}$ | $r_{2}$ | $r_{3}$
$q_{1}$ | $0$ | $q_{1}$ | $0$ | $r_{3}$ | $-r_{2}$ | $-p_{1}$ | $0$ | $0$
$q_{2}$ | $0$ | $q_{2}$ | $-r_{3}$ | $0$ | $r_{1}$ | $0$ | $-p_{1}$ | $0$
$q_{3}$ | $0$ | $q_{3}$ | $r_{2}$ | $-r_{1}$ | $0$ | $0$ | $0$ | $-p_{1}$
$r_{1}$ | $r_{1}$ | $0$ | $-p_{2}$ | $0$ | $0$ | $0$ | $q_{3}$ | $-q_{2}$
$r_{2}$ | $r_{2}$ | $0$ | $0$ | $-p_{2}$ | $0$ | $-q_{3}$ | $0$ | $q_{1}$
$r_{3}$ | $r_{3}$ | $0$ | $0$ | $0$ | $-p_{2}$ | $q_{2}$ | $-q_{1}$ | $0$

######  Example 4.5.

The space spanned by $u=q_{1}$, $v=(p_{1}-p_{2})$ is of type $U$ and the
three-dimensional algebra it generates is spanned as a vector space by
$p_{1},p_{2},q_{1}$ with $\phi_{u,v}(p_{1})=\begin{pmatrix}1&0\\\
0&0\end{pmatrix}$, $\phi_{u,v}(p_{2})=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$,
$\phi_{u,v}(q_{1})=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$

######  Example 4.6.

For $i,j\in\\{1,2,3\\}$, $i\neq j$, the space spanned by $q_{i}$, $r_{j}$ is
of type Z.

###  4.3. A classification of two-dimensional subspaces of
$\operatorname{Im}(O)$

Now that we established the existence of two-dimensional subspaces of
$\operatorname{Im}(O)$ that generate subalgebras that are only 3-dimensional
in case that $O$ is split, it is clear that the proof of Theorem 3.4 does not
immediately extend to fields over which every Cayley algebra is split, such as
$\mathbb{C}$ and $\mathbb{F}_{q}$ (cf Section 2.5). On the other hand it is
also not clear that some small variation of the proof would not still work.
For example while the algebra $\langle p_{1}-p_{2},q_{1}\rangle$ of Ex. 4.5 is
not a simple four-dimensional subalgebra itself, it is certainly contained in
one: the algebra with vector space basis $p_{1},p_{2},q_{1},r_{1}$. We even
have that the map $\phi_{u,v}$ of Ex. 4.5 extend to an isomorphism of this
algebra to $\operatorname{Mat}(2,F)$ by setting
$\phi_{u,v}(r_{1})=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}$. All in all it is
not yet clear that, even when $O$ is split, the set $\mathcal{B}$ of Theorem
3.4 will fail to yield a $q$-Fano plane. In order to trample any optimism
stemming from this observation we classify the remaining two-dimensional
subspaces of $\operatorname{Im}(O)$ and the (four-dimensional) algebras they
generate. Inspired by Propositions 3.3 and 4.2, we base our classification of
two-dimensional subspaces of $\operatorname{Im}(O)$ on the number of nilpotent
lines they contain and in accordance with the famous cliche about the way
mathematicians count, we restict the possibilities for this number to ‘zero’,
‘one’, ‘two’ and ‘all of them’.

######  Definition 4.7.

Let $U$ be a two-dimensional subspace of $\operatorname{Im}(O)$. Then:

  * •

If $U$ contains no nilpotent lines, we say that $U$ is of type Q.

  * •

If $U$ contains exactly one nilpotent line $Fu$ then:

    * –

If $uv\in Fu$ for every $u\in Fu,v\in U$ then we say that $U$ is of type U

    * –

If not, then we say that $U$ is of type D.

  * •

If $U$ contains exactly 2 nilpotent lines, we say that $U$ is of type M.

  * •

If every element in $U$ is nilpotent then:

    * –

If the product of any two elements of $U$ is zero we say that $U$ is of type Z

    * –

If not we say that $U$ is of type J.

In case $O$ is a division algeba, it is clear that only type Q spaces appear
and hence that our classification covers all possible cases. This latter fact
is not yet clear in case $O$ is split, but will be established in Lemma 4.9
below. The names of the types of subspaces are based on the type of
subalgebras of $O$ they generate. The main focus of this section is to show:

######  Theorem 4.8.

Let $U$ be a two-dimensional subspace of $\operatorname{Im}(O)$. Then:

  * •

If $U$ is of type $Q$ then $\langle U\rangle$ is a four-dimensional quaternion
subalgebra of $O$. (Prop. 3.3 above)

  * •

If $U$ is of type U then $\langle U\rangle$ is a three-dimensional subalgebra
of $O$ isomorphic to the upper triangular 2-by-2-matrices (Prop. 4.3 above)

  * •

If $U$ is of type D then $\langle U\rangle$ is a four-dimensional subalgebra
of $O$ with vectorspace decomposition $\langle U\rangle=D\oplus Z$ where $D$
is a two-dimensional Cayley-Dickson subalgebra of $O$, $Z$ is the two-
dimensional Jacobson-radical of $\langle U\rangle$ (so every $z\in Z$ is
nilpotent and $uz\in Z$ for every $z\in Z,u\in\langle U\rangle$) and $Z$
happens to be a type Z subspace of $\operatorname{Im}(O)$. (Prop. 4.11 below.)

  * •

If $U$ is of type $M$ then $\langle U\rangle$ is a four-dimensional split
quaternion subalgebra of $O$, so in particular isomorphic to
$\operatorname{Mat}(2,F)$. (Prop. 4.14 below.)

  * •

If $U$ is of type $Z$ then $\langle U\rangle$ is three-dimensional, with two-
dimensional Jacobson radical $J(\langle U\rangle)=\operatorname{Im}(\langle
U\rangle)=U$ and multiplication as in the text below Proposition 4.2. (Prop.
4.2 above.)

  * •

If $U$ is of type $J$ then $\langle U\rangle$ is a four-dimensional algebra
$F1\oplus J$ with three-dimensional Jacobson radical $J=J(\langle
U\rangle)=\operatorname{Im}(\langle U\rangle)\supsetneq U$. (Prop 4.18 below.)

We note that in order to show that our classification covers all possible
cases, it suffices to show that any two-dimensional subspace $U$ containing
both a non-zero nilpotent $u$ and a non-nilpotent element $v$, is of one of
the three types U, D or M. We will do so by introducing a slightly different
trichotomy for spaces of this type which clearly covers all cases and then
showing that the two trichotomies coincide.

######  Lemma 4.9 (Completeness of the classification of Def. 4.8).

Let $U=\operatorname{span}(u,v)\subset\operatorname{Im}(O)$ with
$u^{2}=0,v^{2}=\alpha 1$ for some $\alpha\neq 0\in F$. We have:

  1. (1)

If $uv\in U$ then $U$ is of type U.

  2. (2)

If $uv\not\in U$ but still $uv\in\operatorname{Im}(O)$ then $U$ is of type D.

  3. (3)

If $uv\not\in\operatorname{Im}(O)$ then $U$ is of type M.

###### Proof.

Since $u,v\in\operatorname{Im}(O)$ we have that $vu=v^{*}u^{*}=(uv)^{*}$ so
that $uv+vu=2\tau(uv)$ by (13). This means that $uv\in\operatorname{Im}(O)$ if
and only if $uv+vu=0$.

When indeed $uv\in\operatorname{Im}(O)$ it follows that $(\lambda u+\mu
v)^{2}=\mu^{2}\alpha$ so that the generic element $(\lambda u+\mu v)$ of $U$
is nilpotent if and only if $\mu=0$. If follows that the only nilpotent
elements of $U$ lie on the line $Fu$. However, since $vu=-uv$ in that case, we
find that $(uv)^{2}=u(vu)v=-u^{2}v^{2}=0$ so that the product $uv$ is
nilpotent. Statements 1 and 2 of the lemma then follow from the definitions of
types U and D.

When $uv\not\in\operatorname{Im}(O)$, we have that $uv+vu=\beta 1$ for some
$\beta\neq 0\in F$. Let $u^{\prime}=\frac{-\alpha}{\beta}u$ Then
$(u^{\prime}+v)^{2}=0+\frac{-\alpha}{\beta}\beta+\alpha=0$. It follows that
$U$ contains at least two distinct nilpotent lines that together span $U$:
$Fu=Fu^{\prime}$ and $F(u^{\prime}+v)$. What remains to be shown is that these
are the only nilpotent lines; that is that $(\lambda
u^{\prime}+\mu(u^{\prime}+v))^{2}=0$ only if $\lambda=0$ or $\mu=0$. To verify
this we compute that $(\lambda
u^{\prime}+\mu(u^{\prime}+v))^{2}=\lambda\mu(u^{\prime}v+vu^{\prime})=\lambda\mu(\frac{-\alpha}{\beta}(uv+vu))=-\lambda\mu\alpha$.
Since $\alpha\neq 0$, the claim follows. ∎

In a curious turn of events the lemma proves its own converse:

######  Corollary 4.10.

  1. (1)

Let $u,v\in\operatorname{Im}(O)$ be elements of a type U space $U$ with
$u^{2}=0$, $v^{2}\neq 0$. Then $uv\in U$.

  2. (2)

Let $u,v\in\operatorname{Im}(O)$ be elements of a type D space $U$ with
$u^{2}=0$, $v^{2}\neq 0$. Then $uv\not\in U$, but still
$uv\in\operatorname{Im}(O)$.

  3. (3)

Let $u,v\in\operatorname{Im}(O)$ be elements of a type M space $U$ with
$u^{2}=0$, $v^{2}\neq 0$. Then $uv\not\in\operatorname{Im}(O)$.

###### Proof.

  1. (1)

If $uv$ landed anywhere else, the space $U$ would be of type $D$ or $M$ by
Lemma 4.9.

  2. (2)

If $uv$ landed anywhere else, the space $U$ would be of type $U$ or $M$ by
Lemma 4.9.

  3. (3)

If $uv$ landed anywhere else (that is: in $\operatorname{Im}(O)$), the space
$U$ would be of type $U$ or $D$ by Lemma 4.9.

∎

This converse is useful as it gives us a description of spaces $U$ of type D
and M that already anticipates the determination of the algebras $\langle
U\rangle$. After all: if $uv\not\in F1\oplus U$ (as is clearly the case for
type D and will later be verified for type M) then Lemma 3.2 tells us that
$\\{1,u,v,uv\\}$ is a basis the space $\langle U\rangle$.

We’ll put this principle into action right away.

###  4.4. The algebras generated by subspaces of type D

######  Proposition 4.11.

Let $U\subseteq\operatorname{Im}(O)$ be two-dimensional of type D, so with
basis $u,v$ satisfying $u^{2}=0$, $v^{2}=\alpha\in F\backslash\\{0\\}$ and
$w\coloneqq uv\in\operatorname{Im}(O)\backslash U$. Then $\langle U\rangle$ is
four-dimensional, with two-dimensional Jacobson radical
$Z=\operatorname{span}(u,w)$ and semi-simple quotient $\langle U\rangle/Z\cong
D_{\alpha}(F)$. The space $Z$ is of type $Z$, all other two-dimensional
subspaces $U^{\prime}$ of $\langle U\rangle$ are either of type $D$ or type
$U$ with the unique nilpotent line in $U^{\prime}$ being the intersection
$U^{\prime}\cap Z$ with the space $Z$.

###### Proof.

Since $u,v,w$ are linearly independent and imaginary it is clear that
$\operatorname{Im}(\langle U\rangle)$ is at least three-dimensional, and by
Lemma 3.2 it is exactly three-dimensional with basis $\\{u,v,w\\}$. Secondly,
we note that $\langle v\rangle=F1\oplus Fv\cong D_{\alpha}(F)$ so that we
obtain a vectorspace decomposition $\langle U\rangle=D_{\alpha}(F)\oplus Z$ so
that, as in text below Theorem 2.49, the claim that $\langle U\rangle/Z\cong
D_{\alpha}(F)$ follows as soon as we establish that $Z$ is indeed the Jacobson
radical. This is easiest from the characterisation of the Jacobson radical in
Lemma 2.51 as the space of all $x\in\langle U\rangle$ satisfying
$xy\in\operatorname{Im}(\langle U\rangle)$ for all $y\in\langle U\rangle$.
Indeed: let $\beta 1+\gamma u+\delta v+\epsilon w$ be an element of the
Jacobson radical, then, setting $y=1$ we find that $\beta=0$ and setting $y=v$
we find that $\delta=0$. This shows that $y\in Z$ so that $J(\langle
U\rangle)\subset Z$. Conversely, since $u1=u$, $u^{2}=0$, $uv=w$ and
$uw=u^{2}v=0$ are all imaginary we have that $u\in J(\langle U\rangle)$ and
similarly imaginarity of $w1=w,wu=-(vu)u=-vu^{2}=0$, $wv=uv^{2}=\alpha u$ and
$w^{2}=-(vu)(uv)=vu^{2}v=0$ implies that $w\in J(\langle U\rangle)$. Together
these inclusions show that $Z\subset J(\langle U\rangle)$. The fact that $Z$
is of type Z follows from the equations $u^{2}=uw=w^{2}=0$ established above.

Let $x=x_{v}+x_{z}$ be the decomposition of $x\in\langle U\rangle$ along the
vectorspace decomposition $\langle U\rangle=\langle v\rangle\oplus Z$. Since
$\langle v\rangle$ is a subalgebra and $Z$ is an ideal we find that $x^{2}=0$
can only hold if $x_{v}^{2}=0$. However since $\langle v\rangle\cong
D_{\alpha}(F)$ which is either a division algebra or isomorphic to the algebra
direct sum $F\oplus F$ (Proposition 2.30) contains no nilpotent elements we
find that $x^{2}=0$ implies that $x_{v}=0$ and hence $x\in Z$. By Lemma 2.45
this means that $Z$ is not only the space of all strongly nilpotent elements
but also the space of _all_ nilpotent elements in $\langle U\rangle$.

With this observation in mind, the claim that all other two-dimensional
subspaces of $\operatorname{Im}(\langle U\rangle)$ are of either type U or
type D is trivial from dimension considerations: since
$\operatorname{Im}(\langle U\rangle)$ is three-dimensional each two-
dimensional subspace $U^{\prime}$ other than $Z$ has one-dimensional
intersection with $Z$ and this intersection consists of all nilpotent elements
in the subspace. Hence $U^{\prime}$ contains exactly one nilpotent line and
thus is of type U or D by Definition 4.7. We will see in Chapter 9 (Corollary
9.20) that type U occurs if and only if $D_{\alpha}(F)$ is split. ∎

######  Example 4.12.

Let $\alpha\in F\backslash\\{0\\}$ and $i\neq j\in\\{1,2,3\\}$. With notation
as in Lemma 4.4 and Table 3, the space spanned by $u=q_{i}$ and
$v=q_{j}-\alpha r_{j}$ is of type D by Lemma 4.9. In particular, since
$v^{2}=\alpha 1$, we see from the proof of Proposition 4.11 that every
isomorphism class of two-dimensional Cayley-Dickson algebras $D_{\alpha}(F)$
appears as the semi-simple part of some subalgebra $\langle U\rangle\subset O$
with $U$ of type D.

###  4.5. The algebras generated by subspaces of type M

For the type M case we work with a slightly different basis (which we can
think of as the pair $(u^{\prime},u^{\prime}+v)$ of the proof of Lemma 4.9):

######  Lemma 4.13.

Let $U\subset\operatorname{Im}(O)$ be a two-dimensional subspace of type M.
Then $U$ has a basis $q,r$ satisfying $q^{2}=r^{2}=0$, $2\tau(qr)=qr+rq=1$.

###### Proof.

Pick non-zero elements $q,r^{\prime}$ on the two nilpotent lines. $q$ and
$r^{\prime}$ span $U$ and $q+r^{\prime}$ can not be a nilpotent which in
particular means that $(q+r^{\prime})^{2}\neq 0$. On the other hand we know
(from $(q+r^{\prime})\in\operatorname{Im}(O)$) that $(q+r^{\prime})^{2}\in
F1$. Define $\alpha\in F$ by $(q+r^{\prime})^{2}=\alpha 1$ and define
$r=r^{\prime}/\alpha$. Then expanding $(q+\alpha r)^{2}=\alpha 1$ yields
$qr+rq=1$. The fact that $qr+rq=2\tau(qr)$ whenever
$q,r\in\operatorname{Im}(O)$ has been derived many times above. ∎

######  Proposition 4.14.

Let $U\subseteq\operatorname{Im}(O)$ be a two-dimensional subspace with basis
elements $q,r$ satisfying $q^{2}=r^{2}=0$, $2\tau(qr)=1$. Then $\langle
U\rangle$ is four-dimensional and isomorphic to $\operatorname{Mat}(2,F)$ with
isomorphism given by

$\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}\mapsto qr;\quad\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\mapsto q;\quad\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}\mapsto
r;\quad\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}\mapsto rq.$

The suggestive notation is chosen partially in order to make it easier to
write down examples of type $M$ subspaces in terms of the basis of Table 3.

###### Proof.

Set $e_{1}=qr$, $e_{2}=rq$. We first verify that $e_{1}$ and $e_{2}$ are
commuting idempotents summing to 1. By (14) we have that
$n(qr)=(qr)^{*}qr=(rq)(qr)=r(qq)r=0$. Hence by $(qr)^{2}=2\tau(qr)qr-n(qr)=qr$
(8) we find idempotence of $qr$. The calculation for $rq$ is identical. We
also notice that $e_{1}e_{2}=qrrq=0$ and $e_{2}e_{1}=rqqr=0$. The identity
$e_{1}+e_{2}=1$ is just the identity $x+x^{*}=2\tau(x)$ applied to $x=qr$.

We see that $\operatorname{span}(e_{1},e_{2})$ is a two-dimensional algebra
and hence
$\operatorname{span}(e_{1},e_{2})\cap\operatorname{Im}(O)=\operatorname{Im}(\operatorname{span}(e_{1},e_{2}))$
is one-dimensional by Lemma 3.1. From $e_{1}^{*}=e_{2}$ it then follows that
$\operatorname{span}(e_{1},e_{2})\cap\operatorname{Im}(O)=F(e_{1}-e_{2})$. In
order to establish linear independence of $e_{1},e_{2},q,r$ it hence suffices
to show that $(e_{1}-e_{2})\not\in U$. Suppose, by contradiction, that $qr-
rq=\lambda q+\mu r$ then, from multiplying on the left by $q$, we would have
that $qrq=-\mu qr$. However, from $(qrq)^{*}=q^{*}r^{*}q^{*}=(-1)^{3}qrq$ we
see that $qrq\in\operatorname{Im}(O)$ so that $\tau(qrq)=0$. On the other hand
$\tau(\mu qr)=\mu/2$ so that $qrq=-\mu qr$ implies that $\mu=0$. Similarly we
arrive at $\lambda=0$ by multiplying our assumption $qr-rq=\lambda q+\mu r$