Problem

We consider$$\sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k a_{ij}(x_{i\nu}-\xi_i)(x_{j\nu}-\xi_j) = \sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k a_{ij}(x_{i\nu}-\bar{x}_i)(x_{j\nu}-\bar{x}_j) - 2n \sum_{i=1}^k \sum_{j=1}^k a_{ij}(\bar{x}_i-\xi_i)(\bar{x}_j-\xi_j),$$where $\bar{x}_i := \frac{1}{n} (x_{i1} + \cdots + x_{in}),\ i = 1,\ldots , k$

My calculate

\begin{align}\sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k a_{ij}(x_{i\nu}-\xi_i)(x_{j\nu}-\xi_j) &=\sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k a_{ij}\{(x_{i\nu}-\bar{x}_i)+(\bar{x}_i-\xi_i)\}\{(x_{j\nu}-\bar{x}_j)+(\bar{x}_j-\xi_j)\}\\ &=\sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k (x_{i\nu}-\bar{x}_i)(x_{j\nu}-\bar{x}_j)+ \underbrace{\sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k (x_{i\nu}-\bar{x}_i)(\bar{x}_j-\xi_j)}_{=0}+ \underbrace{\sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k (\bar{x}_i-\xi_i)(x_{j\nu}-\bar{x}_j)}_{=0}+ \sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k (\bar{x}_i-\xi_i)(\bar{x}_j-\xi_j)\\ &=\sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k (x_{i\nu}-\bar{x}_i)(x_{j\nu}-\bar{x}_j)+ n\sum_{i=1}^k \sum_{j=1}^k (\bar{x}_i-\xi_i)(\bar{x}_j-\xi_j)\end{align}

What is wrong?