## Elements of the coset G/H, where G=$GL^+$(2) and H=SO(2)

In this paper1 in section 2, a method to write the elements of the coset of G/H is provided for GL(4), but I am interested in $GL^+$(2).

My matrix representation of $mathfrak{gl}(2)$ is

$$

begin{bmatrix}

a+x& -b+y\

b+y & a-x

end{bmatrix}

$$

My matrix representation of $mathfrak{so}(2)$ is

$$

begin{bmatrix}

0& -b\

b & 0

end{bmatrix}

$$

Reading the paper, it states that the element g of G can be decomposed as g=$gamma$h, where

$$

gamma = exp left( begin{bmatrix}

a+x& +y\

y & a-x

end{bmatrix} right) in G/H

$$

and where

$$

h = exp left( begin{bmatrix}

0& -b\

b & 0

end{bmatrix} right) in H

$$

Is this correct, or no?

I am suspicious of the argument, because to me

$$

exp left( begin{bmatrix}

a+x& y\

y & a-x

end{bmatrix} right)exp left( begin{bmatrix}

0& -b\

b & 0

end{bmatrix} right) neq exp left( begin{bmatrix}

a+x& -b+y\

b+y & a-x

end{bmatrix} right)

$$

Thus, $gamma h$ does not appear to realize all elements of $G$. Or do we not care about some missing elements for cosets?

$endgroup$