  We provide a unified ultraproduct approach for constructing Wick words in
mixed $q$-Gaussian algebras, which are generated by $s_j=a_j+a_j^*$,
$j=1,\cdots,N$, where $a_ia^*_j - q_{ij}a^*_ja_i =\delta_{ij}$. Here we also
allow equality in $-1\le q_{ij}=q_{ji}\le 1$. Using the ultraproduct method, we
construct an approximate co-multiplication of the mixed $q$-Gaussian algebras.
Based on this we prove that these algebras are weakly amenable and strongly
solid in the sense of Ozawa and Popa. We also encode Speicher's central limit
theorem in the unified ultraproduct method, and show that the
Ornstein--Uhlenbeck semigroup is hypercontractive, the Riesz transform
associated to the number operator is bounded, and the number operator satisfies
the $L_p$ Poincar\'e inequalities with constants $C\sqrt{p}$.
