jemanden in etwas verwickeln
to involve someone in something
To draw someone, often against their will, into a complicated, unpleasant, or dangerous situation.
Er wurde in eine politische Intrige verwickelt.
He was involved in a political intrigue.

to involve / entangle
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to involve someone in something
To draw someone, often against their will, into a complicated, unpleasant, or dangerous situation.
Er wurde in eine politische Intrige verwickelt.
He was involved in a political intrigue.

'Verwickeln' (to involve/entangle) is an inseparable verb. The prefix 'ver-' always remains attached to the stem 'wickeln' (to wrap). It often indicates that something becomes complicated or goes in the wrong direction.
Ich möchte dich nicht in meine Probleme verwickeln.
I don't want to involve you in my problems.
The verb 'verwickeln' (to involve/entangle) is almost always used with the preposition 'in' and the accusative. This applies to all meanings, whether it is about a situation, a conversation, or getting physically caught in something.
Sie war in einen Autounfall verwickelt.
She was involved in a car accident.
You can involve someone in something (transitive) or get yourself entangled/involved in something (reflexive). The reflexive form 'sich verwickeln' (to get entangled/involved) is used when a person or thing gets into a complicated situation by itself.
Der Dieb verwickelte den Verkäufer in ein Gespräch.
The thief engaged the salesperson in a conversation.
When someone unintentionally gets into an unpleasant situation, the passive form 'in etwas verwickelt sein/werden' (to be/become involved in something) is often used. This emphasizes that the person did not act actively.
Er ist in einen großen Finanzskandal verwickelt.
He is involved in a major financial scandal.
The idiom 'sich in Widersprüche verwickeln' (to get caught up in contradictions / to contradict oneself) is a fixed combination. It is used to describe that a person's statements are illogical and do not fit together.
Der Angeklagte verwickelte sich während des Verhörs in Widersprüche.
The defendant contradicted himself during the interrogation.
to participate / involve
'beteiligen' and 'verwickeln' both have to do with someone becoming involved in something or being part of something. The important difference is the evaluation. 'Sich beteiligen' sounds neutral or positive and means that someone actively participates. 'Jemanden in etwas verwickeln' often sounds negative: a person is drawn into a complicated or unpleasant situation. With 'in ein Gespräch verwickeln' (to engage someone in conversation), someone deliberately starts a conversation, often with a specific purpose. Learners may confuse the words because both describe a kind of participation. 'beteiligen' fits voluntary participation, while 'verwickeln' fits more with problems or deliberately drawing someone in.

Der Zeuge wurde in einen langen Streit zwischen den Nachbarn verwickelt.
The witness was drawn into a long dispute between the neighbors.
Lena beteiligt sich morgen an der Diskussion über das neue Essen in der Kantine.
Tomorrow Lena is taking part in the discussion about the new food in the cafeteria.