I have 4 definitions for the following (Einstein summation) tensor
$A^{ijk}A^{*}_{ijk}=A^{111}A^{*}_{111}+3(A^{112}A^{*}_{112})+3(A^{122}A^{*}_{122})+A^{222}A^{*}_{222}$
If I have these 4 definitions and sub them in,
$(A^{111}=a)$, $(A^{112}=b)$, $(A^{122}=c)$, $(A^{222}=d)$
Would the result look like this?
$a^{2}+3b^{2}+3c^{2}+d^{2}$
Let $i,j,k=1,2$. Using Einsten's summaton convention:
$$A^{ijk}B_{ijk}:=A^{1jk}B_{1jk}+A^{2jk}B_{2jk}=\left(A^{11k}B_{11k}+A^{12k}B_{12k} \right)+\left(A^{21k}B_{21k}+A^{22k}B_{22k}\right)=\\ A^{111}B_{111}+A^{112}B_{112}+ A^{121}B_{121}+A^{122}B_{122}+ A^{211}B_{211}+A^{212}B_{212}+ A^{221}B_{221}+A^{222}B_{222}; $$
If both $A$ and $B$ are symmetric in $i,j$, then
$$A^{ijk}B_{ijk}=A^{111}B_{111}+3A^{112}B_{112}+ 3A^{122}B_{122}+A^{222}B_{222}, $$
as you stated. If $B_{\bullet \bullet}:=A^{*}_{\bullet \bullet}$, denoting by $*$ complex conjugation we arrive at
$$A^{ijk}B_{ijk}=|a|^2+3|b|^2+ 3|c|^2+|d|^2, $$
where $$A^{111}=a,$$ $$A^{112}=b,$$ $$A^{122}=c,$$ $$A^{222}=d, $$
and $|\cdot|$ is the modulus of complex numbers.