I am familiar with Yang-Mills equation of motion E.O.M. (without matter or source fields) in differential form.
$$ D * F =0 $$ and Bianchi identity $$ D F=0 $$ where $F= dA + A \wedge A$ and $D=d + [A, ]$ as the covariant derivative version of exterior derivative $d$.
However, in Nakahar book Geometry, Topology and Physics, Second Edition ,
we can compare E.O.M. to his (1.269) below,
and Bianchi identity to his (1.266) below.
My question is that: Did Nakahara make any mistake? Or are his equations the rewriting of my Yang-Mills Equations above? If so, how do we convert to make the rewriting precise?