We propose a unitary diagonalisation of a special class of quaternion matrices, the so-called ηη-Hermitian matrices A = AηH , η∈{ı,J,K} arising in widely linear modelling. In 1915, Autonne exploited the symmetric structure of a matrix A = AT to propose its corresponding factorisation (also known as the Takagi factorisation) in the complex domain C. Similarly, we address the factorisation of an ‘augmented’ class of quaternion matrices, by taking advantage of their structures unique to the quaternion domain H. Applications of such unitary diagonalisation include independent component analysis and convergence analysis in statistical signal processing.