Question: Define a lie group with complex and real variables

I am a beginer in Maple, and even more beginner with differential geometry on maple. I am trying two dayes now to define the following Lie group on Maple by reading the help guid of DifferentialGeometry and some documents on the web, but I failed. Now I am seeking for your assistance, Plese help me and thanks for being as xplicite as you can.

The group I want to introduce is the real lie group

$ \mathbb R \times \mathbb C $ () equipped with the group law
$$ (x_0,x) \cdot (y_0,y) = (x_0+y_0+\frac{1}{2}Im(x\bar{y}), x+y) $$

(I didn't figure out how to use something other than LaTeX code)

For illustration purpose I am sticking with the above simple example, but I want next to move to higher dimensions analog.
I want also an explication about how Maple distinguishes between a real and complex coordinate.

I want next to compute  the left multiplication pushforward of a tangent vector of the identity.
Then compute the structure constants of the Lie algebra associated. I think Maple can do this kind of manipulation without problem, am I wrong?

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