Soda-lime glass is made by melting together soda (sodium carbonate), lime (calcium oxide), dolomite (calcium magnesium carbonate), silica (silicon dioxide), alumina (aluminum oxide), and small quantities of other compounds. Wikipedia says this is done in a furnace at "up to" 1675 °C. Many of the carbonates decompose at lower temperatures, 650 °C for dolomite and 851 °C for soda. However, soda-lime glass begins to soften at 700 °C.
Given the high decomposition temperature of carbonates like soda and dolomite, which is close to and sometimes higher than the temperature at which soda-lime glass begins to soften, one might think that there is a potential for some carbonate to remain dissolved in the glass. This is perhaps supported by the fact that silicon and carbon are both Group IV elements and could share spaces in the same lattice structure.
However, I haven't been able to find a quantitative analysis of soda-lime glass which measures oxidized carbon content. Is this because it's difficult to measure, or because no one in the glass industry cares, or because it's actually very close to zero?