Complexifications in the analytic and algebraic categories

Sanskar, when discussing with me about the definiton of complexification of -Lie groups, came to the following conclusion at 𝔐𝔞𝔵𝔦𝔪𝔞𝔩 ℑ𝔡𝔢𝔞𝔩.

The complexification of as a Lie group is . And complexification of is .

Now consider the -Lie group

Its complexification must be

However, consider the -algebraic group given by

We can write this as

with the identification

with the group law pushed forward.

Now, by construction

However,

Therefore, algebraic complexification and analytic complexification do not agree with each other.