<p>In this paper, we first study an <i>l</i>-linear complementary pair (abbreviated to <i>l</i>-LCP) of codes over the non-chain ring <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(R=\mathbb {F}_q+u\mathbb {F}_q+ v\mathbb {F}_q+uv\mathbb {F}_q\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>=</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>+</mo> <mi>u</mi> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>+</mo> <mi>v</mi> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>+</mo> <mi>u</mi> <mi>v</mi> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(u^2=u,v^2=v, uv=vu\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>=</mo> <mi>u</mi> <mo>,</mo> <msup> <mi>v</mi> <mn>2</mn> </msup> <mo>=</mo> <mi>v</mi> <mo>,</mo> <mi>u</mi> <mi>v</mi> <mo>=</mo> <mi>v</mi> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>. Then we provide a method of constructing asymmetric entanglement-assisted quantum error-correcting (abbreviated to AEAQEC) codes from an <i>l</i>-LCP of codes over <i>R</i> by using CSS. To enrich the variety of available AEAQEC codes, some new AEAQEC codes are given in the sense that their parameters are different from all the previous constructions.</p>