How to Split a String While Keeping Seperators?

I need to do some string splitting for this text compression algorithm, but it must keep the separators.

Pretty simple, say I have a string:

eabdeeceda

I want to split the string by "e"

string.split(str, "e")

Problem is that it removes the separators.

abd, c, da

I want

e, abd, e, e, c, e, da

I've tried to think of various manual methods but they all seem to fall flat in the end.

Here's my best attempt

function splitKeep(str, separator)
    local minimum = 0
    local tbl = {}
    repeat
        local a, b = string.find(str, separator, minimum) -- find first separator
        minimum = b -- set minimum
        local inner
        if string.sub(str, b, b+(b-a)) ~= separator then -- if the next section is not the same as the separator
            local c = string.find(str, separator, minimum) -- find the next separator
            inner = string.sub(str, b, c) -- get the string inbetween the end of the first separator and the start of the second
        end
        tbl[#tbl+1] = string.sub(str, b) -- insert the separator
        tbl[#tbl+1] = inner -- insert the inside string
    until string.find(str, separator, minimum) == nil
    return tbl
end

I already know this wont work, it wont return anything before the first separator and after the last separator. It's also pretty messy.