We propose a unified approach for orthogonal frequency-division multiplexing (OFDM) with grouped subcarriers which encompasses previous schemes such as subcarrier-index modulation (SIM) and grouped linear constellation-precoding (GLCP) as particular cases. To exploit signal space diversity (SSD), we treat the n subcarriers within a group as 2n real resources. An interleaving at the real component level is used to break the correlation between adjacent subcarriers. An error performance analysis then reveals that a 2n-dimensional real signal constellation (codebook) should be designed according to the SSD criterion and the multiplicity-to-product distance ratio criterion. Under certain constraints, the maximum diversity gain of 2n can be achieved by proper design. We design new codebooks based on real lattices and show both through computer simulations and analysis that a superior error performance can be achieved.