$T_8 = T'_{4a}T'_{4b} = ({{{\alpha}^{4}}+{2 i {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}) ({{{\alpha}^{4}}-{2 i {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}) =\\

{{{\alpha}^{8}}+{6 {\alpha}^{4} {\beta}^{4}}}+{{\beta}^{8}}$

and

$T_{12} = \frac {T_{4a}^{'3} + T_{4b}^{'3}}{2} = \frac{{{{{({\alpha}^{4}}+{2 i {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}})^{3}}+{{{{({\alpha}^{4}}-{2 i {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}} )^ {3}}}{2} = \\

{{{{\alpha}^{12}}-{9 {\alpha}^{8} {\beta}^{4}}}-{9 {\alpha}^{4} {\beta}^{8}}}+{{\beta}^{12}}$

while with the missing $\sqrt(3)$ , we get correctly (44) :

$T_8 = T'_{4a}T'_{4b} = ({{{\alpha}^{4}}+{2 i \sqrt{3} {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}) ({{{\alpha}^{4}}-{2 i \sqrt{3} {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}) =\\

{{{\alpha}^{8}}+{14 {\alpha}^{4} {\beta}^{4}}}+{{\beta}^{8}}$

and

$T_{12} = \frac {T_{4\alpha}^{'3} + T_{4\beta}^{'3}}{2} = \frac{{{{{({\alpha}^{4}}+{2 i \sqrt{3} {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}})^{3}}+ {{{{({\alpha}^{4}}-{2 i \sqrt{3} {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}} )^ {3}}}{2} = \\

{{{{\alpha}^{12}}-{33 {\alpha}^{8} {\beta}^{4}}}-{33 {\alpha}^{4} {\beta}^{8}}}+{{\beta}^{12}}$

Idem for the following. I mean the author didn't used the equation with the typo error for the next calculus.

Then it is probably just a well found typo error :) You must write to the author to release a new version.