Proving the 5-Engel identity in the 2-generator group of exponent four

It is known that the fifth Engel word E_5 is trivial in the 2-generator group of exponent four B(2,4), and so can be written as a product of fourth powers. Explicit products of 250 and 28 powers are known, using fourth powers of words up to lengths four and ten respectively. Using a reduction technique based on the recursive enumerability of the set of trivial words in a finite presentation we were able to rewrite E_5 as a product of 26 fourth powers of words up to length five.