Summary: A mixed model \({\mathbf Y}^0= \sum_{i=1}^m {\mathbf X}_i\pmb\beta_i+ \sum_{i=m+1}^w{\mathbf X}_i\widetilde{\pmb\beta}_i+e\) is orthogonal when the matrices \({\mathbf M}_i={\mathbf X}_i{\mathbf X}_i^t\), \(i=1,\dots,w\), commute. The vectors \(\pmb\beta_1^{c_1},\dots, \pmb\beta_m^{c_m}\) are fixed vectors and the \(\pmb\beta_{m+1}^{c_{m+1}},\dots, \pmb\beta_w^{c_w}\) and \textbf{e} are random. We intend to present an approach based on model estimation taking \[ {\mathbf Y}^0={\mathbf L}\bigg( \sum_{i=1}^m {\mathbf X}_i\pmb\beta_i+ \sum_{i=m+1}^w{\mathbf X}_i \widetilde{\pmb\beta}_i\bigg)+{\mathbf e}, \] with \textbf{L} a matrix whose column vectors are linearly independent. The model \({\mathbf Y}^0\) is an extension of the core mixed model. We obtain exact formulae to validate variance component estimators in normal mixed models. An application is also given.