(pp. 37-44)

We continue to assume that

$\Xi \; = \;(K,\;L,\;q,\;1,\;X,\; \cdot ,\;h,\;\theta )$

is a$\delta {\rm{ - }}$standard quadrangular algebra for some$\delta \; \in \;L$(as defined in 4.1) and we continue to let$f,\;\sigma ,\;g,\;\phi $and$\pi $be as in 1.17. In this chapter, we assume as well that

(5.1)${\rm{di}}{{\rm{m}}_K}L\; \le \;4$.

Our goal is to show that under these assumptions,$\Xi $is special (as defined in 1.28).

Proposition 5.2.*Let$a\; \in \;X{\kern 1pt} *$. Then either${\dim _K}L\: = \:2$and$L\; = \;\langle 1,\;\pi (a)\rangle $or${\dim _K}L\: = \:4.$Suppose that${\dim _K}L\: = \:4.$. Then*

(i)$L\; = \;\langle 1,\;\pi (a),\;w,\;\theta (a,\;w)\rangle$and$\langle w,\;\theta (a,\;w)\rangle \; = \;{\langle 1,\;\pi (a)\rangle ^ \bot }$for all non-zero elements$w\; \in \;{\langle 1,\;\pi (a)\rangle ^ \bot }$if either${\text{char}}(K) \ne 2$or${\text{char}}(K)\: = \:2$and$f(\pi (a),\;1)\; \ne \;0$.

(ii)$L\; = \;\langle 1,\;\delta ,\;\pi (a),\;\theta (a,\;\delta )\rangle$and

$\langle \pi (a),\;\theta (a,\;\delta )\rangle \; = \;{\langle 1,\;\delta \rangle ^ \bot }$

if${\text{char}}(K)\: = \:2$and$f(\pi (a),\;1)\; = \;0$.

Proof. By D2,$\langle 1,\;\pi (a)\rangle$is...