In this paper, we consider the identifiability of second-order blind separation of multidimensional components. By maximizing the likelihood for piecewise-stationary Gaussian data, we obtain that the maximum likelihood (ML) solution is equivalent to joint block diagonalization (JBD) of the sample covariance matrices of the observations. Small-error analysis of the solution indicates that the identifiability of the model depends on the positive-definiteness of a matrix, which is a function of the latent source covariance matrices. By analysing this matrix, we derive necessary and sufficient conditions for the model to be identifiable. These are also the sufficient and necessary conditions for JBD of any set of real positive-definite symmetric matrices to be unique.