why is I included as part of GDP? GDP = C + I + G + (X-M)

If GDP is supposed to count only final goods and services, why include I? Eg if build public roads, can count as G; if build factory, can count when factory produces goods and those goods are sold.

Put differently, counting I seems to double count. Eg if one counts spending on factory as I today, when the factory produces and sells goods tomorrow and in subsequent periods, the counting is done again (eg I, followed by C).
Investing is about "new" things that did not exist. Any acquisition of a previously existing asset, such as a house, is considered as Savings rather than Investment. Moreover, Investment is not consumed at the present time and are expected to provide flows for longer periods of time.