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We can think Quaternions as 4-D space over Real numbers,
and 2-D space over Complex numbers as q = (a + b i) + (c + d i)j.
Quaternion is used to describe rotation in 3-D space.
Quaternion with no real part v = b i + c j + d k
is called "vector quaternion", and 3-D space.
For a vector v and a Quaternion q = |q|(cos t/2 + u sin t/2),
q v q^(-1) is a vector v t-rotated along u.
D4 lattice is lattice points of Quaternion q = a + b i + c j + d k as follows.
(1) a,b,c,d are all integer, or
(2) a,b,c,d are all half-integer.
D4 is sub-ring of Quaternion with GCD.
(Ring means a space with +, - and *.)
Octonion is 8-D space over Real number,
4-D space over Complex number,
and 2-D space over Quaternion number as follows.
o=a + b i1 + c i2 + d i3 + e i4 + f i5 + g i6 + h i7
=a + b i + (c + d i) i2 + (e + f i) i4 + (g - h i)i6
=(a + b i1 + c i2 + d i3) + (e + f i1+ g i2 + h i3)i4.
Octonion is used to describe rotation in 7-D space.
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