From 61037c13a8d4988428aeb79b383f3b45dbad8778 Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Mon, 1 Mar 2021 22:32:21 -0500
Subject: [PATCH] octonions: add Cayley-Dickson representation.

---
 mjo/octonions.py | 47 +++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 47 insertions(+)

diff --git a/mjo/octonions.py b/mjo/octonions.py
index f6e222e..cd25f18 100644
--- a/mjo/octonions.py
+++ b/mjo/octonions.py
@@ -1,3 +1,4 @@
+from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
 from sage.combinat.free_module import CombinatorialFreeModule
 from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
 from sage.categories.magmatic_algebras import MagmaticAlgebras
@@ -171,6 +172,52 @@ class Octonion(IndexedFreeModuleElement):
             raise ValueError("zero is not invertible")
         return self.conjugate()/self._norm_squared()
 
+
+    def cayley_dickson(self, Q=None):
+        r"""
+        Return the Cayley-Dickson representation of this element in terms
+        of the quaternion algebra ``Q``.
+
+        The Cayley-Dickson representation is an identification of
+        octionions `x` and `y` with pairs of quaternions `(a,b)` and
+        `(c,d)` respectively such that:
+
+        * `x + y = (a+b, c+d)`
+        * `xy` = (ac - \bar{d}*b, da + b\bar{c})`
+        * `\bar{x} = (a,-b)`
+
+        where `\bar{x}` denotes the conjugate of `x`.
+
+        SETUP::
+
+            sage: from mjo.octonions import Octonions
+
+        EXAMPLES::
+
+            sage: O = Octonions()
+            sage: x = sum(O.gens())
+            sage: x.cayley_dickson()
+            (1 + i + j + k, 1 + i + j + k)
+
+        """
+        if Q is None:
+            Q = QuaternionAlgebra(self.base_ring(), -1, -1)
+
+        i,j,k = Q.gens()
+        a = (self.coefficient(0)*Q.one() +
+             self.coefficient(1)*i +
+             self.coefficient(2)*j +
+             self.coefficient(3)*k )
+        b = (self.coefficient(4)*Q.one() +
+             self.coefficient(5)*i +
+             self.coefficient(6)*j +
+             self.coefficient(7)*k )
+
+        from sage.categories.sets_cat import cartesian_product
+        P = cartesian_product([Q,Q])
+        return P((a,b))
+
+
 class Octonions(CombinatorialFreeModule):
     r"""
     SETUP::
-- 
2.44.2