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The root hyperplanes are $x_1=0$, $x_2=0$, $x_1=x_2$ and $x_1=-x_2$. Their reflections are coordinate sign changes, coordinate interchange and a signed interchange. They generate all signed permutations of the two coordinates:Geometrically this is the dihedral symmetry group of a square, with order eight. It preserves both the four short roots and the four long roots of the B2 root system.

Representatives in $SO(5)$ can be given explicitly. The matrixhas determinant one and conjugates the first rotation block to its inverse, giving $(x_1,x_2)\mapsto (-x_1,x_2)$. The matrix $D_2=\mathrm{diag}(1,1,1,-1,-1)$ similarly reverses $x_2$. Flipping the fifth coordinate compensates for the determinant of the planar reflection.

The permutation matrixswaps the two coordinate planes. Its permutation is $(13)(24)$, so its determinant is also one. Conjugation gives $(x_1,x_2)\mapsto (x_2,x_1)$, the reflection in $x_1=x_2$. These elements normalize $T$, and $P,D_1$ generate all eight Weyl actions. For example, $P{D}_1$ induces $(x_1,x_2)\mapsto (x_2,-x_1)$, a quarter-turn, and $D_1{D}_2$ induces the half-turn. The representatives supply the specified reflections and their typical products inside the actual special orthogonal group.