Geodesic
Representation of binary quadratic forms by a~quaternary form
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 19, Tome 302 (2003), pp. 68-80

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The paper presents formulas for finding the number of representations $r(A;Q)$ of a form $A$ with odd square-free level by the form $Q$ defined by the identity matrix of order 4. The case where the difference of the dimensions $n-m$ is even (the codimension is even), which was not considered previously, is analyzed. 
@article{ZNSL_2003_302_a4,
     author = {N. Yu. Kuranova},
     title = {Representation of binary quadratic forms by a~quaternary form},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {68--80},
     publisher = {mathdoc},
     volume = {302},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_302_a4/}
}
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N. Yu. Kuranova. Representation of binary quadratic forms by a~quaternary form. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 19, Tome 302 (2003), pp. 68-80. http://geodesic.mathdoc.fr/item/ZNSL_2003_302_a4/