is the direct limit of the gl("n"), which embeds in gl("n" + 1) as the upper left block matrix, and formula_28 is its commutator subgroup. define an "elementary matrix" to be one which is the sum of an identity matrix and a single off-diagonal element (this is a subset of the elementary matrices used in linear algebra). then whitehead's lemma states that the group "e"("a") generated by elementary matrices equals the commutator subgroup [gl("a"), gl("a")]. indeed, the group gl("a")/e("a") was first defined and studied by whitehead, and is called the whitehead group of the ring "a".