Today in a talk, it has been mentioned that there exists algebraic groups over the local field $\mathbb{R}$ such that the finite central extension can not be defined algbraically over $\mathbb{R}$ or its algebraic closure $\mathbb{C}$. I guess already $SL(2)$, which is even defined over $\mathbb{Z}$, and the metaplectic group are such an example!?

I am curious, what is the (intuitive) reason for this lack. And, how to proof it rigorously?

Today in a talk, it has been mentioned that there exists algebraic groups over the local field $\mathbb{R}$ such that the finite central extension can not be defined algbraically over $\mathbb{R}$ or its algebraic closure $\mathbb{C}$. I guess already covers of $SL(2)$, which is even defined over $\mathbb{Z}$, and the metaplectic group are such an example!?

I am curious, what is the (intuitive) reason for this lack. And, how to proof it rigorously?

Today in a talk, it has been mentioned that there exists algebraic groups over the local field $\mathbb{R}$ such that the finite central extension can not be defined algbraically over $\mathbb{R}$ or its algebraic closure $\mathbb{C}$. I guess already $SL(2)$, which is even defined over $\mathbb{Z}$, and the metaplectic group are such an example!?

I am curious, what is the reason for this lack.

Today in a talk, it has been mentioned that there exists algebraic groups over the local field $\mathbb{R}$ such that the finite central extension can not be defined algbraically over $\mathbb{R}$ or its algebraic closure $\mathbb{C}$. I guess already $SL(2)$, which is even defined over $\mathbb{Z}$, and the metaplectic group are such an example!?

I am curious, what is the (intuitive) reason for this lack. And, how to proof it rigorously?