The η-inverted ℝ-motivic sphere

We use an Adams spectral sequence to calculate the ℝ-motivic stable homotopy groups after inverting η. The first step is to apply a Bockstein spectral sequence in order to obtain h₁-inverted ℝ-motivic Ext groups, which serve as the input to the η-inverted ℝ-motivic Adams spectral sequence. The second step is to analyze Adams differentials. The final answer is that the Milnor-Witt (4k-1)-stem has order 2^u+1, where u is the 2-adic valuation of 4k. This answer is reminiscent of the classical image of J. We also explore some of the Toda bracket structure of the η-inverted ℝ-motivic stable homotopy groups.