
die Verwicklung
involvement
Participation in, or being drawn into, a complicated, unpleasant, or criminal matter.
Seine Verwicklung in den Finanzskandal kostete ihn seine Karriere.
His involvement in the financial scandal cost him his career.
involvement / complication
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involvement
Participation in, or being drawn into, a complicated, unpleasant, or criminal matter.
Seine Verwicklung in den Finanzskandal kostete ihn seine Karriere.
His involvement in the financial scandal cost him his career.

complication
An unexpected complication or a complex turn of events that makes a situation difficult. Often used in the plural.
Trotz einiger diplomatischer Verwicklungen konnte der Konflikt friedlich gelöst werden.
Despite some diplomatic complications, the conflict was resolved peacefully.
The noun 'Verwicklung' comes from the verb 'verwickeln', which means 'to confuse' or 'to draw into something'. The root 'wickeln' (to wrap) creates the image of being 'wrapped up' in a complicated situation.
Er wollte sich nicht in die Streitigkeiten seiner Kollegen verwickeln lassen.
He did not want to let himself get drawn into his colleagues' disputes.
When the word appears in the singular and often with a possessive pronoun ('seine Verwicklung' — his involvement), it usually refers to personal involvement. In the plural ('die Verwicklungen' — the complications), it more often describes the complicated circumstances themselves.
Seine Verwicklung in die Affäre wurde nie bewiesen.
His involvement in the affair was never proven.
To express what someone is involved in, the preposition 'in' is used with the accusative. Typical phrases are 'Verwicklung in einen Skandal' (involvement in a scandal) or 'in eine Straftat' (in a criminal offense).
Die Polizei untersucht die Verwicklung des Managers in den Betrugsfall.
The police are investigating the manager's involvement in the fraud case.
When 'Verwicklungen' describes complications, it is often combined with adjectives such as 'politisch' (political), 'diplomatisch' (diplomatic), or 'rechtlich' (legal). Another common phrase is 'es kam zu Verwicklungen' (complications arose).
Der Grenzstreit führte zu ernsthaften diplomatischen Verwicklungen.
The border dispute led to serious diplomatic complications.
participation / stake
Beteiligung and Verwicklung can both refer to having a role in a criminal matter. Beteiligung suggests active involvement. Verwicklung can also mean involvement, but leaves open whether someone was drawn into the unpleasant or complicated matter.

Die Polizei prüft seine Verwicklung in den Betrug, weil er möglicherweise ohne Absicht hineingezogen wurde.
The police are investigating his involvement in the fraud because he may have been drawn into it unintentionally.
Die Polizei konnte seine aktive Beteiligung an dem Betrug nachweisen.
The police were able to prove his active involvement in the fraud.