Computer vision (CV) datasets often exhibit biases that are perpetuated by deep learning models. While recent efforts aim to mitigate these biases and foster fair representations, they fail in complex real-world scenarios. In particular, existing methods excel in controlled experiments involving benchmarks with single-attribute injected biases, but struggle with multi-attribute biases being present in well-established CV datasets. Here, we introduce BAdd, a simple yet effective method that allows for learning fair representations invariant to the attributes introducing bias by incorporating features representing these attributes into the backbone. BAdd is evaluated on seven benchmarks and exhibits competitive performance, surpassing state-of-the-art methods on both single- and multi-attribute benchmarks. Notably, BAdd achieves +27.5% and +5.5% absolute accuracy improvements on the challenging multi-attribute benchmarks, FB-Biased-MNIST and CelebA, respectively.