Higher order terms in the non-abelian D-brane effective action and magnetic background fields

Recently a proposal for the bosonic sector of the non-abelian effective D-brane action was given through order alpha'(4). As the resulting expressions turned out to be quite involved, some checks of this result are called for. In the present paper we calculate the spectrum in the presence of constant magnetic background fields and compare it to the string theoretical result. Apart from a small typo in the original expression (the overall sign of the alpha'(4) term), we obtain perfect agreement. We discuss potential applications.