On the existence of Engel pairs in certain linear groups

Let G be a group and h, g is an element of G. The 2-tuple ( h, g) is said to be an n-Engel pair, n = 2, if h = [h,(n) g], g = [ g,(n) h] and h not equal 1. Let SL(2, F) be the special linear group of degree 2 over the field F. In this paper, we show that given any field L, there is a field extension F of L with [ F : L] <= 6 such that SL( 2, F) has an n-Engel pair for some integer n >= 4. We will also show that SL( 2, F) has a 5-Engel pair if F is a field of characteristic p equivalent to +/- 1 mod 5.