I have a 5-dimensional Euclidean space, $E$, with an orthonormal basis ${\bf x}, {\bf y}_{1}, {\bf w}_{1}, {\bf y}_{2}, {\bf w}_{2}$ and two transformations: $$ A_{1} : -{\bf x} + {\bf y}_{1} \mapsto {\bf w}_{1}, \quad {\bf w}_{1} \mapsto {\bf x} - {\bf y}_{1}, \quad {\bf y}_{2} \mapsto {\bf 0}, \quad {\bf w}_{2} \mapsto {\bf 0} $$ and $$ A_{2} : -{\bf x} + {\bf y}_{2} \mapsto {\bf w}_{2}, \quad {\bf w}_{2} \mapsto {\bf x} - {\bf y}_{2}, \quad {\bf y}_{1} \mapsto {\bf 0},\quad {\bf w}_{1} \mapsto {\bf 0}. $$ I am trying to calculate the orbit of a generic point in $E$ under the action of the group, $G_{A_{1}, A_{2}}$, generated by the 1-parameter groups $t \mapsto e^{t A_{1}}, t \mapsto e^{t A_{2}}$.
Here are approaches I have tried and challenges encountered (which are almost surely from my lack of familiarity with the needed theory, so an answer like "read X" is a great answer).
Calculate the Lie algebra generated by $A_{1}, A_{2}$ with the Lie bracket defined as the commutation: $[A_{1}, A_{2}] = A_{2} A_{1} - A_{1} A_{2}$. I am getting that the dimension is no lower than 6. But even having a basis for the Lie algebra, is there a theory for determining the group $G_{A_{1}, A_{2}}$? (The underlying field is the reals, not algebraically closed, so Dynkin diagrams would not apply.)
Trying to consider, first, the two 1-parameter groups as if acting on two disjoint 3-D Euclidean spaces (then the orbit of a point is the torus $S^{1} \times S^{1}$), and then trying to calculate the topological factor space $(S^{1} \times S^{1}) / \sim$ by a suitable relation $\sim$ (which identifies two 1-dimensional subspaces from each 3-D space), but have had a hard time "seeing" the factor space.
Anyway, if there is a theory, I'd appreciate some sources. If I am missing something straightforward, being pointed toward it would be great.