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I came across a synthesis of centrohexaindane that has the following steps:

Centrohexaindane synthesis

This is a Diels–Alder reaction that gives an aromatic ring. I know that alkynes will give this reaction as follows:

Diels–Alder

But in the scheme, after the initial Diels-Alder to form the product with bridging $\ce{SO2},$ it cannot just lose $\ce{SO2}$ to become aromatic. Does the molecule with the bridging $\ce{SO2}$ get oxidized to have two double bonds like in the second image and then lose $\ce{SO2}?$

The sulfone is the only possible oxidizing agent here (toluene solvent doesn't seem like it would oxidize) and Wikipedia says on its page on sulfones that:

Sulfone is a relatively inert functional group, being weakly basic(compared to sulfoxides). They are non-oxidizing.

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A few references into the rabbit hole and we have an answer. The major reference was given in the link mentioned by OP which was Chem. Ber. 1994, 127 (2), 409–425[1], in which the reaction mechanism of the synthesis is given as follows:

enter image description here

[...] Reaction of 33 with tetrathiophene-S,S-dioxide is performed by heating the reactants in highly concentrated toluene solution to $\pu{120 °C}$ for a total of 24 h. The cycloadduct 34 is obtained in high yield.

Now the question that has arisen in the step from 33 to 34 is the following, under normal conditions of Diels-Alder, there would only be one double bond in the final product. Here there are two. This can be answered going one down into the rabbit hole and using J. Org. Chem. 1980, 45 (5), 856–867.[2]

This paper deals with the specific reaction that causes confusion - the annulation using tetrachlorothiophene 1,1-dioxide. According to this paper:

The compound has a combination of stability and reactivity that makes it a useful synthetic reagent. It undergoes Diels-Alder addition with double bonds to form an adduct which spontaneously loses sulfur dioxide to leave the double bond annelated with a tetrachlorobutadienediyl group.

enter image description here

Furthermore, it talks about the reactivity to olefinic bonds and states the following:

enter image description here

Example 18, from l-hexen-5-yne, reveals the greater reactivity of alkene vs. alkyne. Reaction is exclusively with the alkene bond as far as NMR reveals. Thus, the kinetically favored product is formed rather than the thermodynamically favored one.

There is a minor error in the reaction mechanism as given in the link of OP as the first paper shows that the tetra-chloro derivative needs to be oxidised in order to be converted into cyclohexaindane

References:

  1. Kuck, D. Benzoanellated Centropolyquinanes, 15. Benzoanellated Fenestranes with [5.5.5], [5.5.5.6], and [5.5.5.5] Frameworks: The Route from 1,3-Indandione to Fenestrindan. Chem. Ber. 1994, 127 (2), 409–425. DOI: 10.1002/cber.19941270218.
  2. Raasch, M. S. Annulations with tetrachlorothiophene 1,1-dioxide. J. Org. Chem. 1980, 45 (5), 856–867. DOI: 10.1021/jo01293a019.
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