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Let $G$ be a connected, semisimple linear algebraic group over a field $k$. In Springer's Linear Algebraic group, the Fundamental Group of $G$ is defined by a certain quotient of groups $P/Q$, which is defined via the root system $(X,R)$ of $G(k^{alg})$ (Springer, 8.1.11). Here $k^{alg}|k$ is an algebraic closure, $$P:=\{v\in X\otimes \mathbb{R}~|~\langle v,R^\vee\rangle\subset\mathbb{Z}\}$$ is the weight lattice and $$Q\subset X$$ is the subgroup generated by $R$, the root lattice. In Milne's Algebraic Groups the Fundamental Group of $G$ is defined as the kernel of the universal/simply connected covering $G_{sc}\rightarrow G$ (Milne(online version), 20.5). Now in (Springer, 17.3.1), it is said that the root system corresponding to the Dynkin-classes $D_n$ have Fundamental Group with order $4$ ($n\ge 4$)). But also in (Springer, 17.3.1) for $char(k)\neq 2$, a $k$-split simple group in class $D_n$ is given by (some) $SO_{2n}$. In Serre's Galois Cohomology (III §3.2 b)) the universal cover for $SO_{2n}$ ($n\ge 2$) is given by $$0\rightarrow\mu_2\rightarrow Spin_{2n}\rightarrow SO_{2n}\rightarrow 0,$$ so the (group of $k$-valued or $k^{alg}$-valued points of the) Fundamental Group by Milne has order $2$.

What is the connection between these notions? Or am I missunderstanding something?

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If $(G,T)$ is a split reductive group, then $X^*(Z))=X^*(T)/\mathbb{Z}\Phi$, where $Z$ is the center of $G$ and $\mathbb{Z}\Phi$ the submodule generated by the roots (Milne 2017, 21.6). When $G$ is a simply connected semisimple group, this becomes $X^*(Z)=P/Q$ (ibid. 24c); in this case, $G$ is the universal covering group of the adjoint group $G/Z$ and $Z$ is the fundamental group of $G/Z$.

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