# Quaternion Basics

Algebra Level 3

Let $$A$$, $$B$$ and $$C$$ be quaternions whose values are

$A = 2 - i + j - k$

$B = 3 + i - j + k$

$C = 1 + i + j + k$

with $$i$$, $$j$$, and $$k$$ as the quaternion units following the property

$i^2 = j^2 = k^2 = ijk = -1$

If $$Q = \large \sum\limits_{cyc} ABC$$ and is expressed in the form $$w + xi + yj + zk$$, with $$w$$, $$x$$, $$y$$, and $$z$$ being integers, find $$w+x+y+z$$.

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