[math] \newcommand{\mathds}{\mathbb}[/math]

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Find axioms for the projective quadruplets

[[math]] \xymatrix@R=50pt@C=50pt{ P\ar[r]\ar[d]\ar[dr]&PT\ar[l]\ar[d]\ar[dl]\\ PU\ar[u]\ar[ur]\ar[r]&PK\ar[l]\ar[ul]\ar[u] } [[/math]]

covering the real, complex and free quadruplets, constructed above.