Let $G$ be a simply connected semisimple group over a perfect field $k$ (at the moment I am interested in the case $k=\mathbb R$).
Then $G$ is an inner form of a quasi-split $k$-group $G\_{\rm qs}$:
there exists a quasi-split form $G\_{\rm qs}$ of $G$ and a 1-cocycle $c\in Z^1(k,\overline{G}\_{\rm qs})$ such that
$G=\,\_c G\_{\rm qs}$, the inner twist of $G\_{\rm qs}$ by the cocycle $c$, where $\overline{G}\_{\rm qs}=G\_{\rm qs}/Z\_{\rm qs}$ and $Z\_{\rm qs}=Z(G\_{\rm qs})$.
Let $\xi=[c]\in H^1(k,\overline{G}\_{\rm qs})$, the cohomology class of the cocycle $c$.
Let
$$\Delta\colon H^1(k,\overline{G}\_{\rm qs})\to H^2(k,Z\_{\rm qs})$$
denote the connecting map from the cohomology exact sequence
$$H^1(k,Z\_{\rm qs})\to H^1(k,G\_{\rm qs})\to H^1(k, \overline{G}\_{\rm qs})\to H^2(k,Z\_{\rm qs})$$
induced by the short exact sequence
$$ 1\to Z\_{\rm qs}\to G\_{\rm qs}\to \overline{G}\_{\rm qs}\to 1.$$
By definition, the *Tits class of $G$* is
$$t\_G=\Delta(\xi)\in H^2(k,Z\_{\rm qs}),$$
see The Book of Involutions, (31.7).
Note that $Z\_{\rm qs}=Z:=Z(G)$.



> 
> **Question 1.** Is there anywhere in the literature or in the Internet a table of $t\_G$ for all simply connected absolutely simple ${\mathbb R}$-groups $G$?
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For all connected Dynkin diagram except for ${\mathsf D}\_{2m}$ (and also for ${\mathsf D}\_{2m}$ when $G\_{\rm qs}$ is not split)
we have either $H^2({\mathbb R},Z\_{\rm qs})=0$ or $H^2({\mathbb R},Z\_{\rm qs})={\mathbb Z}/2{\mathbb Z}$.
From the cohomology exact sequence we see that $t\_G=0$ if and only if $G$ is a *strong inner* form of $G\_{\rm qs}$, that is, $\xi$ comes from $H^1({\mathbb R}, G)$.
I have a table of strong inner forms of $G\_{\rm qs}$, and so I know when $t\_G=0$; for non-strong inner forms we have $t\_G\neq 0$, hence $t\_G=1+2{\mathbb Z}\in {\mathbb Z}/2{\mathbb Z}$.


However, for ${\mathsf D}\_{2m}$, in the case when $G\_{\rm qs}$ is split, namely $G\_{\rm qs}\simeq {\bf Spin}(2m,2m)$, we have $H^2({\mathbb R}, Z\_{\rm qs})={\mathbb Z}/2{\mathbb Z}\times{\mathbb Z}/2{\mathbb Z}$.
I know that $t\_G=0$ if and only if $G$ is a strong inner form of $G\_{\rm qs}$, that is, if $G\simeq{\bf Spin}(2m+4q, 2m-4q)$,
but I need an explicit formula for the cases ${\bf Spin}(2m+4q+2, 2m-4q-2)$ and the quaternionic form ${\bf Spin}^\*(4m)$.
Thus Question 1 reduces to the following question:



> 
> **Question 2.** What are the Tits classes for the simple ${\mathbb R}$-groups ${\bf Spin}(2m+4q+2, 2m-4q-2)$ and ${\bf Spin}^\*(4m)$ ?
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In order to formulate an answer to Question 2, one needs an explicit description of the center of the split ${\mathbb R}$-group ${\bf Spin}(2m,2m)$.
Such a description is given, for example, in [Brian Conrad's cheat sheet](http://www.math.stonybrook.edu/~jstarr/papers/Centers.pdf).



The Tits classes of simply connected simple real groups
**Question 1:** I haven't seen an explicit table in the literature of the Tits classes for simple $R$-groups. 


That said, such a table can be constructed from tables in the literature. Specifically, the Tits class is determined by the Tits algebras corresponding to the minuscule dominant weights by Proposition 7 in my paper *Outer automorphisms of algebraic groups and determining groups by their maximal tori* 
(Michigan Mathematical Journal 61 #2 (2012), 227-237).


These Tits algebras are described for each simple type in general terms for any $k$ in section 27 of *The Book of Involutions*, and their precise values for $k = \mathbb{R}$ are available in several references, for example Tits's *Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen*.


**Question 2:** Victor answered this question in the language of Tits algebras, not the Tits class, so I just translate his answer. Identify the projections on the two components of $H^2(\mathbb{R}, Z\_{qs}) \cong \mathbb{Z}/2 \times \mathbb{Z}/2$ with the highest weights of the half-spin representations as in section 4.3 of Tits's paper *Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque*; this agrees with the identification of $Z\_{qs}$ with $\mu\_2 \times \mu\_2$ that you refer to. For $SO(2m + 4q + 2, 2m - 4q - 2)$, both half-spin representations are quaternionic, so the Tits class is $(1,1)$. For $SO^\*(4m)$, one half-spin representation is quaternionic and the other is real, so the Tits class is $(0,1)$ or $(1,0)$.


