$\left. {\begin{array}{*{20}{c}}
{{S_{a + b}} = \frac{{3\sqrt 3 }}{2}{{(a + b)}^2}} \\
{{S_{a - b}} = \frac{{3\sqrt 3 }}{2}{{(a + b)}^2}}
\end{array}} \right\} \to {S_{a + b}} + {S_{a - b}} = \frac{{3\sqrt 3 }}{2}({(a + b)^2} + {(a - b)^2}) = \frac{{3\sqrt 3 }}{2}({a^2} + {b^2} + \cancel{{2ab}} + {a^2} + {b^2} - \cancel{{2ab}})$
$ = \frac{{3\sqrt 3 }}{2}(2{a^2} + 2{b^2})$
$ = \frac{{3\sqrt 3 }}{2} \times 2{a^2} + \frac{{3\sqrt 3 }}{2} \times 2{a^2} = 2\underbrace {\left( {\frac{{3\sqrt 3 }}{2}{a^2}} \right)}_{{S_a}} + 2\underbrace {\left( {\frac{{3\sqrt 3 }}{2}{b^2}} \right)}_{{S_b}}$
$ = 2{S_a} + 2{S_b}$