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May 11, 2015 at 22:13 vote accept Boaz Tsaban
May 11, 2015 at 21:08 comment added Will Sawin @BoazTsaban My algebra, with David Speyer's modification, is spanned by a group. Indeed the set of matrices that are equal to the identity plus one of the elements of my original algebra are a group.
May 11, 2015 at 20:20 comment added David E Speyer And, if "algebra" means "containing the identity" for the OP, then just take the span of Will's example together with the identity matrix.
May 11, 2015 at 20:01 history answered Will Sawin CC BY-SA 3.0