Your question uses a confusing terminology, which is not your fault (it's confused in the literature too).
1. The word "elastomer" stands for a polymer network with no solvent (in fact elastomer is synonymous to "rubber"). When a crosslinked polymer network is swollen in a solvent, this has to be called a "gel".
2. "Lyotropic" liquid crystal phase is the one (different from "thermotropic") where the order is established due to a changing concentration of solvent (rather than temperature change). This actually almost always means that it's the packing effects of anisotropic excluded volume, that result in an LC phase, but this is not your question. Some liquid crystalline polymer systems really are lyotropic - but also many thermotropic liquid crystalline polymers will become isotropic if swollen with a solvent (as any liquid crystal will, when mixed with a lot of isotropic impurity).
3. There is no contradiction in terms (except the use of "elastomer"). You can have a lyotropic liquid crystalline system crosslinked into a network (thus becoming a "gel"!), and still respond to changes in solvent concentration by changing phases.
Your question uses a confusing terminology, which is not your fault (it's confused in the literature too).
1. The word "elastomer" stands for a polymer network with no solvent (in fact elastomer is synonymous to "rubber"). When a crosslinked polymer network is swollen in a solvent, this has to be called a "gel".
2. "Lyotropic" liquid crystal phase is the one (different from "thermotropic") where the order is established due to a changing concentration of solvent (rather than temperature change). This actually almost always means that it's the packing effects of anisotropic excluded volume, that result in an LC phase, but this is not your question. Some liquid crystalline polymer systems really are lyotropic - but also many thermotropic liquid crystalline polymers will become isotropic if swollen with a solvent (as any liquid crystal will, when mixed with a lot of isotropic impurity).
3. There is no contradiction in terms (except the use of "elastomer"). You can have a lyotropic liquid crystalline system crosslinked into a network (thus becoming a "gel"!), and still respond to changes in solvent concentration by changing phases.