@Ben: The "simplest" argument that gives some bound may be one of those already mentioned, unless somebody finds a really new approach. Jordan's result is impressively general in scope, but the tools relevant to proving it seem quite limited. A more transparent proof would obviously be welcome. But after Frobenius and Schur, the published treatments (Curtis-Reiner, Isaacs,...) are basically similar apart from the way they are integrated with other theorems on finite groups and linear groups. Leaving aside the question of best bounds, it's hard to find a fresh approach.

Your initial formulation of an "equivalence" has no role for $\lambda$ on the left side, so needs some modification. It helps here to cite a particular source.

This is a helpful reference to an out-of-the-way paper, shifting the focus from compact Lie groups to general compact groups. For the published version: MR1950883 (2004c:20069) 20F65 (20E18 20G20) Bass, Hyman (1-MI); Lubotzky, Alexander (IL-HEBR); Magid, Andy R. (1-OK); Mozes, Shahar (IL-HEBR), The proalgebraic completion of rigid groups. Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000). Geom.Dedicata 95 (2002), 19–58. This seems to be the same as the preprint, but slightly corrected.

Dixmier is certainly a useful reference (though Bourbaki is also available in English). One caution: the North-Holland translation of Dixmier (translator unidentified) was republished without change by AMS in 1996 but listed some misprints at the end; in fact this doesn't include all of the misprints introduced in the translation. I encouraged the AMS to acquire this out-of-print book for the GSM series but wasn't aware then of the extent of misprints. On the other hand, Dixmier added some updates to the English version.

This reinforces the fact that the notion of "reductive" for a Lie algebra in characteristic 0 has no intrinsic interest, unless you study the Lie algebra of a Lie (or algebraic) group and relate their representations carefully. For affine algebraic groups in any characteristic the notion of "reductive" group is more interesting because of Chevalley's Jordan decomposition and its preservation under rational representations. (But in characteristic $p$ you lose the connection with complete reducibility: a reductive group is "geometrically" reductive but rarely "linearly" reductive.)