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You can't tell by looking at the outside. There are four different limited slip carriers that could go in that housing. BUT, if you are certain it is a 3.08:1 gear set, it has to be a 671 carrier. You can find info about such at https://supercarsunlimited.com/info/...id/diff_id.htm.
p.s. it looks like you better attach your gas tank ground strap.
That is an O-Type rearend and as stated above there are multiple carriers depending upon the gear ratio. It is a unique Oldsmobile-only 8.5” unit that is not interchangeable with the common 8.5” corporate rears.
These rears don’t have a lot of aftermarket support so don’t expect to find parts at Summit, Jegs, etc. The good thing is someone figured out how to machine the Ford 8.8 posi carrier for the O-Type 2.93-3.23 gear sets and those are available from multiple sources. SupercarsUnlimited has parts, as does monzaz on this site - he has a differential business and can likely get what you need.
I changed my '69 3.23:1 "O" type differential from open to limited slip through SupercarsUnlimited.com. As I recall, the owner there (Greg Rollin; he will answer the phone (503-641-1442) and talk with you about it; he's very Oldsmobile-knowledgeable) has conversions for the 3.08:1 "O" type as well. Expect it to cost around $1000 for all the parts.