[Altered State]

The treatment of lactic acid with SOCl2 gives a C6H8O4 product.

I need to draw the structure and the mechanism that gives the product.

I know that any carb. acid with SOCl2 undergoes An + E reaction and gives the acid chloride. I tried, but I cannot figure out a way of drawing a mechanism in which the product is one of that formula.
I guess 2 of the starting molecules would combine in any step, but I cannot see how.
Any thoughts?

[opsomath]

I guess 2 of the starting molecules would combine in any step, but I cannot see how.

I know that any carb. acid with SOCl2 undergoes An + E reaction and gives the acid chloride.

So say you make the acid chloride. What reaction can happen then?

[Babcock_Hall]

@OP, In general into what other classes of compounds can acid chlorides be converted?

[Altered State]

I think I got it:

Lactic acid reacts with SOCl to give the corresponding acid chloride.
Then 2 moleculoes of that cholride acid can reac, each of then with it's OH group acting as nucleophile  and undergoing firstly a nucleophilc additon intermolecular, and then, the other OH group can reat with the other carbonyl undergoing another An, givig as result a cyclic double ester, "double lactone".

[opsomath]

Yep. That's a pretty good description.