Cartan-decomposition subgroups of 𝑆𝑂(2,𝑛)

For G = SL ⁡ ( 3 , R ) G = \operatorname {SL} (3,\mathbb {R}) and G = SO ⁡ ( 2 , n ) G = \operatorname {SO}(2,n) , we give explicit, practical conditions that determine whether or not a closed, connected subgroup H H of G G has the property that there exists a compact subset C C of G G with C H C = G CHC = G . To do this, we fix a Cartan decomposition G = K A + K G = K A^+ K of G G , and then carry out an approximate calculation of ( K H K ) ∩ A + (KHK) \cap A^+ for each closed, connected subgroup H H of G G .