This calculator uses the Adem relations to express elements of the mod 2 Steenrod algebra in terms of the Serre-Cartan basis.
To evaluate (for example) Sq²Sq⁷+Sq⁹, use input format: 2 7 + 9
Currently this is the only input format supported, i.e. input must be a single Steenrod square, a product of squares, or a sum of products of squares - no parenthesization allowed.
Output is in the same format.

Written by Alex Kruckman.
Contact: akruckman at wesleyan dot edu
Source code: adem.rkt
Compiled to Javascript from Racket (a Scheme dialect) using Whalesong.

I wrote the app above for a fun programming diversion during my first semester of graduate school. Since many people have found it useful, I will host it here as long as my Berkeley webspace continues to exist. But I would like to advertise the existence of a Sage package for computations in the Steenrod algebra, which is much more flexible than my simple calculator. Full documentation is here.

For example, the following Sage code expresses Sq²Sq⁷+Sq⁹ in terms of the Serre-Cartan basis in the mod 2 Steenrod algebra.
A = SteenrodAlgebra(basis="serre-cartan")
A.Sq(2)*A.Sq(7)+A.Sq(9)


Sage can handle the mod p Steenrod algebras too. The following code expresses P¹ β P³ in terms of the Serre-Cartan basis in the mod 7 Steenrod algebra. Note that Q(0) represents the Bockstein operation β.
A = SteenrodAlgebra(7,basis="serre-cartan")
A.P(1)*A.Q(0)*A.P(3)

Try it in your browser on SageMathCell.