We propose an expansion of multivariate time-series data into maximally independent source subspaces. The search is made among rotations of prewhitened data which maximize non-Gaussianity of candidate sources. We use a tensorial invariant approximation of the multivariate negentropy in terms of a linear combination of squared coskewness and cokurtosis. By solving a high-order singular value decomposition problem, we extract the axes associated with most non-Gaussianity. Moreover, an estimate of the Gaussian subspace is provided by the trailing singular vectors. The independent subspaces are obtained through the search of quasi-independent components within the estimated non-Gaussian subspace, followed by the identification of groups with significant joint negentropies. Sources result essentially from the coherency of extremes of the data components. The method is then applied to the global sea surface temperature anomalies, equatorward of 65 degrees, after being tested with non-Gaussian surrogates consistent with the data anomalies. The main emerging independent components and subspaces, supposedly generated by independent forcing, include different variability modes, namely, The East-Pacific, the Central Pacific, and the Atlantic Ninos, the Atlantic Multidecadal Oscillation, along with the subtropical dipoles in the Indian, South Pacific, and South-Atlantic oceans. Benefits and usefulness of independent subspaces are then discussed.