$K(p) = k_1 k_2 + \overline{k_1} \overline{k_2}$, where $K(p)$ is the curvature of a riemannian surface

Question

Let $M$ be riemannian $2$-dimensional manifold with curvature $K$ and let $\psi : M \to {\mathbb{R}}^4$ be a isometric immersion. Let $p \in M$ and $\{\eta , \xi\} \subset {\mathbb{R}}^4$ an ortonormal basis in ${(T_pM)}^{\bot}$. Let $k_1$ and $k_2$ be principal curvatures and $\overline{k_1}$ and $\overline{k_2}$ associated to the Weingarten endomorphisms $A_{\eta}$ and $A_{\xi}$. Then $$ K(p) = k_1 k_2+\overline{k_1} \overline{k_2} $$ using the Gauss equation. On the other hand, I know that $a_{1 1} a_{2 2} - a_{1 2}^2 = \det(A{\eta}) = k_1k_2$ (and the same for $\det(A{\xi})$), using that the second fundamental form $II_p(e_i , e_j) = a_{i j}\eta+ \overline{a_{ij}}\xi$ if $\{e_1 , e_2\}$ is a ortonormal basis in $T_pM$