This page lists some examples of Adem relations, sorted by degree. The general formula is that when $i \lt 2j$, $\mathrm{Sq}^i \mathrm{Sq}^j = \sum_{k \in \mathbb{Z}} \binom{j - k - 1}{i - 2k} \mathrm{Sq}^{i + j - k} \mathrm{Sq}^k,$ with the understanding that $\binom{n}{m} = 0$ if $m \gt n$ or $m \lt 0$, and $\mathrm{Sq}^k = 0$ if $k \lt 0$ (so in practice $k$ ranges from $\max(0, i - j + 1)$ to $\lfloor \frac{i}{2}\rfloor$).
I learnt the following "derivation rule" from Eric Peterson: all the Adem relations follow from $\mathrm{Sq}^{2n - 1} \mathrm{Sq}^n = 0$ and asserting that the map $\mathrm{Sq}^{n} \mapsto \mathrm{Sq}^{n - 1}$ is a derivation.
$\mathrm{Sq}^{1}\mathrm{Sq}^{1} = 0$ $\mathrm{Sq}^{1}\mathrm{Sq}^{2} = \mathrm{Sq}^{3}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{3} = 0$ $\mathrm{Sq}^{2}\mathrm{Sq}^{2} = \mathrm{Sq}^{3}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{4} = \mathrm{Sq}^{5}$ $\mathrm{Sq}^{2}\mathrm{Sq}^{3} = \mathrm{Sq}^{5} + \mathrm{Sq}^{4}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{2} = 0$ $\mathrm{Sq}^{1}\mathrm{Sq}^{5} = 0$ $\mathrm{Sq}^{2}\mathrm{Sq}^{4} = \mathrm{Sq}^{6} + \mathrm{Sq}^{5}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{3} = \mathrm{Sq}^{5}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{6} = \mathrm{Sq}^{7}$ $\mathrm{Sq}^{2}\mathrm{Sq}^{5} = \mathrm{Sq}^{6}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{4} = \mathrm{Sq}^{7}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{3} = \mathrm{Sq}^{5}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{7} = 0$ $\mathrm{Sq}^{2}\mathrm{Sq}^{6} = \mathrm{Sq}^{7}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{5} = \mathrm{Sq}^{7}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{4} = \mathrm{Sq}^{7}\mathrm{Sq}^{1} + \mathrm{Sq}^{6}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{3} = 0$ $\mathrm{Sq}^{1}\mathrm{Sq}^{8} = \mathrm{Sq}^{9}$ $\mathrm{Sq}^{2}\mathrm{Sq}^{7} = \mathrm{Sq}^{9} + \mathrm{Sq}^{8}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{6} = 0$ $\mathrm{Sq}^{4}\mathrm{Sq}^{5} = \mathrm{Sq}^{9} + \mathrm{Sq}^{8}\mathrm{Sq}^{1} + \mathrm{Sq}^{7}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{4} = \mathrm{Sq}^{7}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{9} = 0$ $\mathrm{Sq}^{2}\mathrm{Sq}^{8} = \mathrm{Sq}^{10} + \mathrm{Sq}^{9}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{7} = \mathrm{Sq}^{9}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{6} = \mathrm{Sq}^{10} + \mathrm{Sq}^{8}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{5} = \mathrm{Sq}^{9}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{6}\mathrm{Sq}^{4} = \mathrm{Sq}^{7}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{10} = \mathrm{Sq}^{11}$ $\mathrm{Sq}^{2}\mathrm{Sq}^{9} = \mathrm{Sq}^{10}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{8} = \mathrm{Sq}^{11}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{7} = \mathrm{Sq}^{11} + \mathrm{Sq}^{9}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{6} = \mathrm{Sq}^{11} + \mathrm{Sq}^{9}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{6}\mathrm{Sq}^{5} = \mathrm{Sq}^{9}\mathrm{Sq}^{2} + \mathrm{Sq}^{8}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{4} = 0$ $\mathrm{Sq}^{1}\mathrm{Sq}^{11} = 0$ $\mathrm{Sq}^{2}\mathrm{Sq}^{10} = \mathrm{Sq}^{11}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{9} = \mathrm{Sq}^{11}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{8} = \mathrm{Sq}^{12} + \mathrm{Sq}^{11}\mathrm{Sq}^{1} + \mathrm{Sq}^{10}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{7} = 0$ $\mathrm{Sq}^{6}\mathrm{Sq}^{6} = \mathrm{Sq}^{11}\mathrm{Sq}^{1} + \mathrm{Sq}^{10}\mathrm{Sq}^{2} + \mathrm{Sq}^{9}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{5} = \mathrm{Sq}^{9}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{12} = \mathrm{Sq}^{13}$ $\mathrm{Sq}^{2}\mathrm{Sq}^{11} = \mathrm{Sq}^{13} + \mathrm{Sq}^{12}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{10} = 0$ $\mathrm{Sq}^{4}\mathrm{Sq}^{9} = \mathrm{Sq}^{12}\mathrm{Sq}^{1} + \mathrm{Sq}^{11}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{8} = \mathrm{Sq}^{13} + \mathrm{Sq}^{11}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{6}\mathrm{Sq}^{7} = \mathrm{Sq}^{13} + \mathrm{Sq}^{12}\mathrm{Sq}^{1} + \mathrm{Sq}^{10}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{6} = \mathrm{Sq}^{11}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{8}\mathrm{Sq}^{5} = \mathrm{Sq}^{9}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{13} = 0$ $\mathrm{Sq}^{2}\mathrm{Sq}^{12} = \mathrm{Sq}^{14} + \mathrm{Sq}^{13}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{11} = \mathrm{Sq}^{13}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{10} = \mathrm{Sq}^{12}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{9} = \mathrm{Sq}^{13}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{6}\mathrm{Sq}^{8} = \mathrm{Sq}^{14} + \mathrm{Sq}^{13}\mathrm{Sq}^{1} + \mathrm{Sq}^{11}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{7} = \mathrm{Sq}^{13}\mathrm{Sq}^{1} + \mathrm{Sq}^{11}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{8}\mathrm{Sq}^{6} = \mathrm{Sq}^{11}\mathrm{Sq}^{3} + \mathrm{Sq}^{10}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{9}\mathrm{Sq}^{5} = 0$ $\mathrm{Sq}^{1}\mathrm{Sq}^{14} = \mathrm{Sq}^{15}$ $\mathrm{Sq}^{2}\mathrm{Sq}^{13} = \mathrm{Sq}^{14}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{12} = \mathrm{Sq}^{15}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{11} = \mathrm{Sq}^{13}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{10} = \mathrm{Sq}^{13}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{6}\mathrm{Sq}^{9} = \mathrm{Sq}^{14}\mathrm{Sq}^{1} + \mathrm{Sq}^{13}\mathrm{Sq}^{2} + \mathrm{Sq}^{12}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}$ $\mathrm{Sq}^{8}\mathrm{Sq}^{7} = \mathrm{Sq}^{13}\mathrm{Sq}^{2} + \mathrm{Sq}^{12}\mathrm{Sq}^{3} + \mathrm{Sq}^{11}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{9}\mathrm{Sq}^{6} = \mathrm{Sq}^{11}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{15} = 0$ $\mathrm{Sq}^{2}\mathrm{Sq}^{14} = \mathrm{Sq}^{15}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{13} = \mathrm{Sq}^{15}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{12} = \mathrm{Sq}^{15}\mathrm{Sq}^{1} + \mathrm{Sq}^{14}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{11} = 0$ $\mathrm{Sq}^{6}\mathrm{Sq}^{10} = \mathrm{Sq}^{14}\mathrm{Sq}^{2} + \mathrm{Sq}^{13}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{9} = \mathrm{Sq}^{15}\mathrm{Sq}^{1} + \mathrm{Sq}^{13}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{8}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{1} + \mathrm{Sq}^{14}\mathrm{Sq}^{2} + \mathrm{Sq}^{12}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{9}\mathrm{Sq}^{7} = \mathrm{Sq}^{13}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{10}\mathrm{Sq}^{6} = \mathrm{Sq}^{11}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{16} = \mathrm{Sq}^{17}$ $\mathrm{Sq}^{2}\mathrm{Sq}^{15} = \mathrm{Sq}^{17} + \mathrm{Sq}^{16}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{14} = 0$ $\mathrm{Sq}^{4}\mathrm{Sq}^{13} = \mathrm{Sq}^{17} + \mathrm{Sq}^{16}\mathrm{Sq}^{1} + \mathrm{Sq}^{15}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{12} = \mathrm{Sq}^{15}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{6}\mathrm{Sq}^{11} = \mathrm{Sq}^{14}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{10} = \mathrm{Sq}^{15}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{8}\mathrm{Sq}^{9} = \mathrm{Sq}^{17} + \mathrm{Sq}^{16}\mathrm{Sq}^{1} + \mathrm{Sq}^{15}\mathrm{Sq}^{2} + \mathrm{Sq}^{13}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{9}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{2} + \mathrm{Sq}^{13}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{10}\mathrm{Sq}^{7} = \mathrm{Sq}^{13}\mathrm{Sq}^{4} + \mathrm{Sq}^{12}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{11}\mathrm{Sq}^{6} = 0$ $\mathrm{Sq}^{1}\mathrm{Sq}^{17} = 0$ $\mathrm{Sq}^{2}\mathrm{Sq}^{16} = \mathrm{Sq}^{18} + \mathrm{Sq}^{17}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{15} = \mathrm{Sq}^{17}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{14} = \mathrm{Sq}^{18} + \mathrm{Sq}^{16}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{13} = \mathrm{Sq}^{17}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{6}\mathrm{Sq}^{12} = \mathrm{Sq}^{15}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{11} = \mathrm{Sq}^{15}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{8}\mathrm{Sq}^{10} = \mathrm{Sq}^{18} + \mathrm{Sq}^{16}\mathrm{Sq}^{2} + \mathrm{Sq}^{15}\mathrm{Sq}^{3} + \mathrm{Sq}^{14}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{9}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{10}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{3} + \mathrm{Sq}^{14}\mathrm{Sq}^{4} + \mathrm{Sq}^{13}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{11}\mathrm{Sq}^{7} = \mathrm{Sq}^{13}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{18} = \mathrm{Sq}^{19}$ $\mathrm{Sq}^{2}\mathrm{Sq}^{17} = \mathrm{Sq}^{18}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{16} = \mathrm{Sq}^{19}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{15} = \mathrm{Sq}^{19} + \mathrm{Sq}^{17}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{14} = \mathrm{Sq}^{19} + \mathrm{Sq}^{17}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{6}\mathrm{Sq}^{13} = \mathrm{Sq}^{17}\mathrm{Sq}^{2} + \mathrm{Sq}^{16}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{12} = 0$ $\mathrm{Sq}^{8}\mathrm{Sq}^{11} = \mathrm{Sq}^{19} + \mathrm{Sq}^{16}\mathrm{Sq}^{3} + \mathrm{Sq}^{15}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{9}\mathrm{Sq}^{10} = \mathrm{Sq}^{19} + \mathrm{Sq}^{17}\mathrm{Sq}^{2} + \mathrm{Sq}^{15}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{10}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{2} + \mathrm{Sq}^{16}\mathrm{Sq}^{3} + \mathrm{Sq}^{14}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{11}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{12}\mathrm{Sq}^{7} = \mathrm{Sq}^{13}\mathrm{Sq}^{6}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{19} = 0$ $\mathrm{Sq}^{2}\mathrm{Sq}^{18} = \mathrm{Sq}^{19}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{17} = \mathrm{Sq}^{19}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{16} = \mathrm{Sq}^{20} + \mathrm{Sq}^{19}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{15} = 0$ $\mathrm{Sq}^{6}\mathrm{Sq}^{14} = \mathrm{Sq}^{19}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{2} + \mathrm{Sq}^{17}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{13} = \mathrm{Sq}^{17}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{8}\mathrm{Sq}^{12} = \mathrm{Sq}^{20} + \mathrm{Sq}^{16}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{9}\mathrm{Sq}^{11} = \mathrm{Sq}^{17}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{10}\mathrm{Sq}^{10} = \mathrm{Sq}^{19}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{2} + \mathrm{Sq}^{17}\mathrm{Sq}^{3} + \mathrm{Sq}^{15}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{11}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{3} + \mathrm{Sq}^{15}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{12}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{5} + \mathrm{Sq}^{14}\mathrm{Sq}^{6}$ $\mathrm{Sq}^{13}\mathrm{Sq}^{7} = 0$ $\mathrm{Sq}^{1}\mathrm{Sq}^{20} = \mathrm{Sq}^{21}$ $\mathrm{Sq}^{2}\mathrm{Sq}^{19} = \mathrm{Sq}^{21} + \mathrm{Sq}^{20}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{18} = 0$ $\mathrm{Sq}^{4}\mathrm{Sq}^{17} = \mathrm{Sq}^{20}\mathrm{Sq}^{1} + \mathrm{Sq}^{19}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{16} = \mathrm{Sq}^{21} + \mathrm{Sq}^{19}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{6}\mathrm{Sq}^{15} = \mathrm{Sq}^{21} + \mathrm{Sq}^{20}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{14} = \mathrm{Sq}^{19}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{8}\mathrm{Sq}^{13} = \mathrm{Sq}^{21} + \mathrm{Sq}^{17}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{9}\mathrm{Sq}^{12} = \mathrm{Sq}^{21} + \mathrm{Sq}^{17}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{10}\mathrm{Sq}^{11} = \mathrm{Sq}^{21} + \mathrm{Sq}^{20}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{3} + \mathrm{Sq}^{17}\mathrm{Sq}^{4} + \mathrm{Sq}^{16}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{11}\mathrm{Sq}^{10} = \mathrm{Sq}^{19}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{12}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{4} + \mathrm{Sq}^{16}\mathrm{Sq}^{5} + \mathrm{Sq}^{15}\mathrm{Sq}^{6}$ $\mathrm{Sq}^{13}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{6}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{21} = 0$ $\mathrm{Sq}^{2}\mathrm{Sq}^{20} = \mathrm{Sq}^{22} + \mathrm{Sq}^{21}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{19} = \mathrm{Sq}^{21}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{18} = \mathrm{Sq}^{20}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{17} = \mathrm{Sq}^{21}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{6}\mathrm{Sq}^{16} = \mathrm{Sq}^{22} + \mathrm{Sq}^{21}\mathrm{Sq}^{1} + \mathrm{Sq}^{19}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{15} = \mathrm{Sq}^{21}\mathrm{Sq}^{1} + \mathrm{Sq}^{19}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{8}\mathrm{Sq}^{14} = \mathrm{Sq}^{22} + \mathrm{Sq}^{19}\mathrm{Sq}^{3} + \mathrm{Sq}^{18}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{9}\mathrm{Sq}^{13} = 0$ $\mathrm{Sq}^{10}\mathrm{Sq}^{12} = \mathrm{Sq}^{22} + \mathrm{Sq}^{21}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{4} + \mathrm{Sq}^{17}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{11}\mathrm{Sq}^{11} = \mathrm{Sq}^{21}\mathrm{Sq}^{1} + \mathrm{Sq}^{19}\mathrm{Sq}^{3} + \mathrm{Sq}^{17}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{12}\mathrm{Sq}^{10} = \mathrm{Sq}^{19}\mathrm{Sq}^{3} + \mathrm{Sq}^{18}\mathrm{Sq}^{4} + \mathrm{Sq}^{16}\mathrm{Sq}^{6}$ $\mathrm{Sq}^{13}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{14}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{7}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{22} = \mathrm{Sq}^{23}$ $\mathrm{Sq}^{2}\mathrm{Sq}^{21} = \mathrm{Sq}^{22}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{20} = \mathrm{Sq}^{23}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{19} = \mathrm{Sq}^{21}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{18} = \mathrm{Sq}^{21}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{6}\mathrm{Sq}^{17} = \mathrm{Sq}^{22}\mathrm{Sq}^{1} + \mathrm{Sq}^{21}\mathrm{Sq}^{2} + \mathrm{Sq}^{20}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{16} = \mathrm{Sq}^{23}$ $\mathrm{Sq}^{8}\mathrm{Sq}^{15} = \mathrm{Sq}^{23} + \mathrm{Sq}^{21}\mathrm{Sq}^{2} + \mathrm{Sq}^{20}\mathrm{Sq}^{3} + \mathrm{Sq}^{19}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{9}\mathrm{Sq}^{14} = \mathrm{Sq}^{23} + \mathrm{Sq}^{19}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{10}\mathrm{Sq}^{13} = \mathrm{Sq}^{22}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{11}\mathrm{Sq}^{12} = \mathrm{Sq}^{23} + \mathrm{Sq}^{19}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{12}\mathrm{Sq}^{11} = \mathrm{Sq}^{21}\mathrm{Sq}^{2} + \mathrm{Sq}^{20}\mathrm{Sq}^{3} + \mathrm{Sq}^{19}\mathrm{Sq}^{4} + \mathrm{Sq}^{17}\mathrm{Sq}^{6}$ $\mathrm{Sq}^{13}\mathrm{Sq}^{10} = \mathrm{Sq}^{19}\mathrm{Sq}^{4} + \mathrm{Sq}^{17}\mathrm{Sq}^{6}$ $\mathrm{Sq}^{14}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{6} + \mathrm{Sq}^{16}\mathrm{Sq}^{7}$ $\mathrm{Sq}^{15}\mathrm{Sq}^{8} = 0$ $\mathrm{Sq}^{1}\mathrm{Sq}^{23} = 0$ $\mathrm{Sq}^{2}\mathrm{Sq}^{22} = \mathrm{Sq}^{23}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{21} = \mathrm{Sq}^{23}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{4}\mathrm{Sq}^{20} = \mathrm{Sq}^{23}\mathrm{Sq}^{1} + \mathrm{Sq}^{22}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{19} = 0$ $\mathrm{Sq}^{6}\mathrm{Sq}^{18} = \mathrm{Sq}^{22}\mathrm{Sq}^{2} + \mathrm{Sq}^{21}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{17} = \mathrm{Sq}^{23}\mathrm{Sq}^{1} + \mathrm{Sq}^{21}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{8}\mathrm{Sq}^{16} = \mathrm{Sq}^{24} + \mathrm{Sq}^{23}\mathrm{Sq}^{1} + \mathrm{Sq}^{22}\mathrm{Sq}^{2} + \mathrm{Sq}^{20}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{9}\mathrm{Sq}^{15} = \mathrm{Sq}^{21}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{10}\mathrm{Sq}^{14} = \mathrm{Sq}^{23}\mathrm{Sq}^{1} + \mathrm{Sq}^{19}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{11}\mathrm{Sq}^{13} = \mathrm{Sq}^{23}\mathrm{Sq}^{1} + \mathrm{Sq}^{19}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{12}\mathrm{Sq}^{12} = \mathrm{Sq}^{23}\mathrm{Sq}^{1} + \mathrm{Sq}^{22}\mathrm{Sq}^{2} + \mathrm{Sq}^{20}\mathrm{Sq}^{4} + \mathrm{Sq}^{19}\mathrm{Sq}^{5} + \mathrm{Sq}^{18}\mathrm{Sq}^{6}$ $\mathrm{Sq}^{13}\mathrm{Sq}^{11} = \mathrm{Sq}^{21}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{14}\mathrm{Sq}^{10} = \mathrm{Sq}^{19}\mathrm{Sq}^{5} + \mathrm{Sq}^{18}\mathrm{Sq}^{6} + \mathrm{Sq}^{17}\mathrm{Sq}^{7}$ $\mathrm{Sq}^{15}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{7}$ $\mathrm{Sq}^{1}\mathrm{Sq}^{24} = \mathrm{Sq}^{25}$ $\mathrm{Sq}^{2}\mathrm{Sq}^{23} = \mathrm{Sq}^{25} + \mathrm{Sq}^{24}\mathrm{Sq}^{1}$ $\mathrm{Sq}^{3}\mathrm{Sq}^{22} = 0$ $\mathrm{Sq}^{4}\mathrm{Sq}^{21} = \mathrm{Sq}^{25} + \mathrm{Sq}^{24}\mathrm{Sq}^{1} + \mathrm{Sq}^{23}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{5}\mathrm{Sq}^{20} = \mathrm{Sq}^{23}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{6}\mathrm{Sq}^{19} = \mathrm{Sq}^{22}\mathrm{Sq}^{3}$ $\mathrm{Sq}^{7}\mathrm{Sq}^{18} = \mathrm{Sq}^{23}\mathrm{Sq}^{2}$ $\mathrm{Sq}^{8}\mathrm{Sq}^{17} = \mathrm{Sq}^{24}\mathrm{Sq}^{1} + \mathrm{Sq}^{23}\mathrm{Sq}^{2} + \mathrm{Sq}^{21}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{9}\mathrm{Sq}^{16} = \mathrm{Sq}^{25} + \mathrm{Sq}^{23}\mathrm{Sq}^{2} + \mathrm{Sq}^{21}\mathrm{Sq}^{4}$ $\mathrm{Sq}^{10}\mathrm{Sq}^{15} = \mathrm{Sq}^{25} + \mathrm{Sq}^{24}\mathrm{Sq}^{1} + \mathrm{Sq}^{21}\mathrm{Sq}^{4} + \mathrm{Sq}^{20}\mathrm{Sq}^{5}$ $\mathrm{Sq}^{11}\mathrm{Sq}^{14} = 0$ $\mathrm{Sq}^{12}\mathrm{Sq}^{13} = \mathrm{Sq}^{25} + \mathrm{Sq}^{24}\mathrm{Sq}^{1} + \mathrm{Sq}^{23}\mathrm{Sq}^{2} + \mathrm{Sq}^{20}\mathrm{Sq}^{5} + \mathrm{Sq}^{19}\mathrm{Sq}^{6}$ $\mathrm{Sq}^{13}\mathrm{Sq}^{12} = \mathrm{Sq}^{23}\mathrm{Sq}^{2} + \mathrm{Sq}^{21}\mathrm{Sq}^{4} + \mathrm{Sq}^{19}\mathrm{Sq}^{6}$ $\mathrm{Sq}^{14}\mathrm{Sq}^{11} = \mathrm{Sq}^{21}\mathrm{Sq}^{4} + \mathrm{Sq}^{20}\mathrm{Sq}^{5} + \mathrm{Sq}^{18}\mathrm{Sq}^{7}$ $\mathrm{Sq}^{15}\mathrm{Sq}^{10} = \mathrm{Sq}^{19}\mathrm{Sq}^{6}$ $\mathrm{Sq}^{16}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{8}$