Large language models (LLMs) are increasingly used as computational models of human language processing, yet it remains unclear which components of their representations drive alignment with neural activity. Here, we use ridge reduced-rank regression to identify the brain-predictive subspace of LLM representations: the representational directions that support linear prediction of neural responses. Across three naturalistic language-listening datasets spanning intracranial electrophysiology and fMRI, and six model conditions spanning autoregressive, bidirectional masked, and randomly initialized language models, we find that full linear encoding performance can be recovered using only 1-3% of the original embedding dimensionality. These low-dimensional subspaces generalize across held-out participants and, to a lesser extent, across datasets, suggesting that neural predictivity is concentrated in a compact and partially shared representational space. We next compare these brain-predictive directions with the principal axes of variation in the embeddings. Principal component analysis requires substantially more dimensions to recover neural prediction, demonstrating that the directions most useful for predicting neural activity are distinct from those capturing the most overall variance within LLM representations themselves. Finally, characterizing the learned subspaces allows us to surface interpretable features that are overrepresented in neural activity above and beyond the LLM representations. Together, these results show that LLM-brain alignment is supported by a small, structured subset of model representations and provides a framework for better understanding exactly why these representations align. Code can be found at: https://github.com/thefinnlab/low-rank-alignment

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