diff --git a/lib/OrdinaryDiffEqExponentialRK/src/alg_utils.jl b/lib/OrdinaryDiffEqExponentialRK/src/alg_utils.jl
index 1079cb0ca22..e77d3b72c94 100644
--- a/lib/OrdinaryDiffEqExponentialRK/src/alg_utils.jl
+++ b/lib/OrdinaryDiffEqExponentialRK/src/alg_utils.jl
@@ -26,6 +26,37 @@ alg_order(alg::Exprb43) = 4
 alg_adaptive_order(alg::Exprb32) = 2
 alg_adaptive_order(alg::Exprb43) = 4
 
+function _expRK_has_concretization(A)
+    return try
+        has_concretization(A)
+    catch err
+        err isa UndefVarError || rethrow()
+        true
+    end
+end
+
+_expRK_requires_krylov(f) = false
+function _expRK_requires_krylov(f::SplitFunction)
+    A = f.f1.f
+    return size(A) != () && !_expRK_has_concretization(A)
+end
+
+for Alg in (LawsonEuler, NorsettEuler, ETDRK2, ETDRK3, ETDRK4, HochOst4)
+    @eval function DiffEqBase.prepare_alg(
+            alg::$Alg,
+            u0::AbstractArray,
+            p, prob
+        )
+        if !alg.krylov && _expRK_requires_krylov(prob.f)
+            return $Alg(
+                krylov = true, m = alg.m, iop = alg.iop,
+                autodiff = alg.autodiff, concrete_jac = alg.concrete_jac
+            )
+        end
+        return alg
+    end
+end
+
 function DiffEqBase.prepare_alg(
         alg::ETD2,
         u0::AbstractArray,
diff --git a/lib/OrdinaryDiffEqExponentialRK/src/exponential_rk_caches.jl b/lib/OrdinaryDiffEqExponentialRK/src/exponential_rk_caches.jl
index 359819a1664..7515a97288a 100644
--- a/lib/OrdinaryDiffEqExponentialRK/src/exponential_rk_caches.jl
+++ b/lib/OrdinaryDiffEqExponentialRK/src/exponential_rk_caches.jl
@@ -61,6 +61,8 @@ function expRK_operators(::HochOst4, dt, A)
     return A21, A31, A32, A41, A42, A51, A52, A54, B1, B4, B5
 end
 
+_expRK_matrix_or_scalar(A) = size(A) == () ? convert(Number, A) : convert(AbstractMatrix, A)
+
 # Unified constructor for constant caches
 for (Alg, Cache) in [
         (:LawsonEuler, :LawsonEulerConstantCache),
@@ -89,8 +91,7 @@ for (Alg, Cache) in [
         else
             isa(f, SplitFunction) ||
                 throw(ArgumentError("Caching can only be used with SplitFunction"))
-            A = size(f.f1.f) == () ? convert(Number, f.f1.f) :
-                convert(AbstractMatrix, f.f1.f)
+            A = _expRK_matrix_or_scalar(f.f1.f)
             ops = expRK_operators(alg, dt, A)
         end
         if isa(f, SplitFunction) || SciMLBase.has_jac(f)
@@ -143,7 +144,7 @@ function alg_cache_expRK(
     else
         KsCache = nothing
         # Precompute the operators
-        A = size(f.f1.f) == () ? convert(Number, f.f1.f) : convert(AbstractMatrix, f.f1.f)
+        A = _expRK_matrix_or_scalar(f.f1.f)
         ops = expRK_operators(alg, dt, A)
     end
     return uf, jac_config, J, ops, KsCache
@@ -202,7 +203,7 @@ function alg_cache(
         KsCache = (Ks, expv_cache)
     else
         KsCache = nothing
-        A = size(f.f1.f) == () ? convert(Number, f.f1.f) : convert(AbstractMatrix, f.f1.f)
+        A = _expRK_matrix_or_scalar(f.f1.f)
         exphA = expRK_operators(alg, dt, A)
     end
     return LawsonEulerCache(u, uprev, tmp, dz, rtmp, G, du1, jac_config, uf, J, exphA, KsCache)
diff --git a/lib/OrdinaryDiffEqExponentialRK/test/linear_nonlinear_convergence_tests.jl b/lib/OrdinaryDiffEqExponentialRK/test/linear_nonlinear_convergence_tests.jl
index 64c94cc606d..ab8486b06e9 100644
--- a/lib/OrdinaryDiffEqExponentialRK/test/linear_nonlinear_convergence_tests.jl
+++ b/lib/OrdinaryDiffEqExponentialRK/test/linear_nonlinear_convergence_tests.jl
@@ -1,6 +1,7 @@
 using OrdinaryDiffEqExponentialRK, Test, DiffEqDevTools, Random, LinearAlgebra, LinearSolve
 using OrdinaryDiffEqVerner, OrdinaryDiffEqSDIRK
 using OrdinaryDiffEqCore: alg_order
+using SciMLBase: successful_retcode
 
 @testset "Caching Out-of-place" begin
     println("Caching Out-of-place")
@@ -73,6 +74,27 @@ end
     @test sim.𝒪est[:L2] ≈ 4 atol = 0.1
 end
 
+@testset "Matrix-free SciMLOperator split" begin
+    u0 = ComplexF64[1.0 + 0.5im, -0.5 + 0.25im, 0.75 - 0.125im, -0.25 - 0.5im]
+    λ = ComplexF64[-1.0, -2.0, -3.0, -4.0]
+    F = FunctionOperator(
+        (v, u, p, t) -> v, u0;
+        T = ComplexF64,
+        islinear = true,
+        op_inverse = (v, u, p, t) -> v,
+        opnorm = (_ -> 1.0)
+    )
+    L = cache_operator(F \ DiagonalOperator(λ) * F, u0)
+    @test !has_concretization(L)
+
+    prob = SplitODEProblem(L, (u, p, t) -> zero(u), u0, (0.0, 0.1))
+    sol = solve(prob, ETDRK4(), dt = 0.01, save_everystep = false)
+
+    @test successful_retcode(sol)
+    @test sol.alg.krylov
+    @test sol.u[end]≈exp.(0.1 .* λ) .* u0 rtol=1.0e-6
+end
+
 @info "CFNLIRK3() is broken"
 
 @testset "EPIRK Out-of-place" begin