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Suppose $$\mathcal {S}$$S is a finite near hexagon of order (s, t) having v points. For every $$i \in \{ 0,1,2,3\}$$i∈{0,1,2,3}, let $$A_i$$Ai denote the adjacency matrix of the graph defined on the points by the distance i relation. We perform a study of the real algebra generated by the $$A_i$$Ai’s, and take a closer look to the structure of these algebras for all known examples of $$\mathcal {S}$$S. Among other things, we show that a certain number $$d_\mathcal {S}$$dS (which is a function of s, t and v) must be integral. This allows us to exclude certain near hexagons whose (non)existence was already open for about 15 years. In the special case $$s=2$$s=2, we also show that the embedding rank of the near hexagon is at least the number $$d_{\mathcal {S}}$$dS, and that the near hexagon has non-full projective dimensions with vector dimension equal to $$d_{\mathcal {S}}$$dS.

On the adjacency algebras of near hexagons with an order